Conformal structures and solitons in pseudo-Riemannian geometry

INTERNATIONAL DOCTORAL SCHOOL OF THE USC María Ferreiro Subrido PhD Thesis Conformal structures and solitons in pseudo-Riemannian geometry Santiago de Compostela, 2023 Doctoral Programme in Mathematics TESE DE DOUTORAMENTO Conformal structures and solitons in pseudo-Riemannian geometry María Ferreiro Subrido ESCOLA DE DOUTORAMENTO INTERNACIONAL DA UNIVERSIDADE DE SANTIAGO DE COMPOSTELA PROGRAMA DE DOUTORAMENTO EN MATEMÁTICAS SANTIAGO DE COMPOSTELA ANO 2023 . DECLARACIÓN DO AUTOR DA TESE Conformal structures and solitons in pseudo-Riemannian geometry María Ferreiro Subrido Presento a miña tese seguindo o procedemento adecuado ao Regulamento e declaro que: 1) A tese abarca os resultados da elaboración do meu traballo. 2) No seu caso, na tese faise referencia ás colaboracións que tivo este traballo. 3) A tese é a versión definitiva presentada para a súa defensa e coincide coa versión enviada en formato electrónico. 4) Confirmo que a tese non incorre en ningún tipo de plaxio doutra autoría nin de traballos presentados por min para a obtención doutros títulos. En Santiago de Compostela, 27 de abril de 2023 Asdo. María Ferreiro Subrido . AUTORIZACIÓN DOS DIRECTORES DA TESE Conformal structures and solitons in pseudo-Riemannian geometry D. Eduardo García Río e D. Ramón Vázquez Lorenzo INFORMAN: Que a presente tese se corresponde co traballo realizado por María Ferreiro Subrido baixo a nosa dirección e autorizamos a súa presentación, considerando que reúne os requisitos esixidos no Regulamento de Estudos de Doutoramento da USC, e que como directores desta non incorremos nas causas de abstención establecidas na lei 40/2015. En Santiago de Compostela, 27 de abril de 2023 Asdo. Eduardo García Río Asdo. Ramón Vázquez Lorenzo . Os resultados presentes nesta memoria foron obtidos coa axuda do financiamento da Consellería de Cultura, Educación e Ordenación Universitaria da Xunta de Galicia, na modalidade de Grupo de Referencia Competitiva: GRC2013-045. Agradecemos tamén a axuda dos proxectos MTM2016-75897-P, ED431F2017/03 e ED431F 2020/04 (co cofinanciamento do FEDER) e da convocatoria de axudas predoutorais FPU19/00130. . Resumo Sexa (M, g)unha variedade pseudo-Riemanniana e denotemos por [g]a clase das métricas pseudo-Riemannianas que difiren de gnunha transformación conforme. Un problema básico en xeometría Riemanniana e conforme é o estudo da existencia de estruturas distinguidas (como poden ser as métricas de Einstein, as estruturas Kähler ou para-Kähler, os solitóns de Ricci, etc.) na clase conforme dunha métrica dada. No marco desta tese de doutoramento examinamos este tipo de problemas obtendo resultados de clasificación baixo certas condicións adicionais. A existencia de métricas chás na clase conforme dunha métrica dada (é dicir, a existencia de métricas conformemente chás) é unha propiedade moi restritiva. Dende o punto de vista local, en dimensión catro esta cuestión vén caracterizada pola anulación do tensor de curvatura de Weyl, o que garantiza a existencia local de funcións σ:U ⊂ M→Rtales que (U, g =e2σg)é chá. Se ben existen abundantes exemplos de métricas localmente conformemente chás, a esfera Snproporciona unha mostra de que tal condición non pode ser sempre estendida a unha condición global. O noso primeiro obxectivo foi a determinación da estrutura local das variedades Kählerianas e para-Kählerianas que conteñen algunha representante chá na súa clase conforme. En dimensión maior que catro xa se coñecía que esta situación só pode darse sobre variedades chás, pero a dimensión catro aínda constituía un problema aberto. Patterson mostrou en [119] a existencia de exemplos de variedades Kählerianas non chás con operador de Ricci nilpotente en dous pasos no caso de signatura neutra. No Capítulo 2 desta memoria proporcionamos a descrición da estrutura local destas variedades obtendo a existencia de exemplos con operador de Ricci complexo. Teorema 2.2 Toda superficie Kähleriana localmente conformemente chá indescompoñible (M, g, J+)é localmente isométrica ao fibrado cotanxente (T∗Σ, g)dunha superficie Riemanniana (Σ, gΣ)de curvatura constante coa métrica dada por unha das dúas seguintes posibilidades. (i) g=g∇Σse a curvatura de Gauss e distinta de cero, ou (ii) g=ιJΣ◦ιId +g∇Σse a curvatura de Gauss é idénticamente cero, xv onde ∇Σdenota a conexión de Levi-Civita de (Σ, gΣ)eJΣé a estrutura Kähleriana en Σasociada á forma de volume Riemanniana. Ademais, e estrutura complexa J+ en T∗Σvén determinada pola forma simpléctica Ω+=−dιJΣ. Un resultado en certo modo análogo describe as superficies para-Kählerianas que son localmente conformemente chás. Teorema 2.1 Toda superficie para-Kähleriana localmente conformemente chá indescompoñible (M, g, J−)é localmente isométrica ao fibrado cotanxente T∗Σdunha superficie afín chá (Σ, D)coa estrutura para-complexa determinada por J|ker π∗= Id, onde πdenota a proxección canónica do fibrado cotanxente e a métrica está dada pola estensión de Riemann modificada g=ιT ◦ιId +gD, onde (i) Té un campo de tensores de tipo (1,1) nilpotente e paralelo sobre (Σ, D), ou (ii) Té un campo de tensores de tipo (1,1) paralelo sobre (Σ, D)que satisfai T2=−κ2Id. Ademais, en ambos casos a dúas-forma de Kähler Ω−(X, Y ) = g(J−X, Y )é a forma simpléctica canónica do fibrado cotanxente. Se ben en ambos casos as estruturas locais se corresponden con estensións de Riemann modificadas, as conexións na variedade base son esencialmente diferentes. No caso Kähleriano estamos a estender a conexión de Levi-Civita dunha superficie con curvatura de Gauss constante, mentres que na situación para-Kähleriana estendemos unha conexión afín chá; o cal proporciona un maior grao de liberdade con respecto á situación complexa. O operador de Ricci correspondente ás métricas nas afirmacións (ii) dos Teoremas 2.1 e 2.2 determinan unha estrutura complexa paralela auto-adxunta sobre a variedade, o cal dá lugar a unha estrutura anti-Kähler (ver [21]) cuxa métrica comparte a conexión de Levi-Civita coa métrica de partida. Dado que as métricas dos Teoremas 2.1 e 2.2 son localmente simétricas, cuestionámonos a posibilidade da súa realización como métricas invariantes á esquerda en grupos de Lie. Ovando determinara as estruturas Kählerianas, tanto en signatura Riemanniana como neutra, sobre grupos de Lie de dimensión catro en [116]. Porén, a descrición dos grupos de Lie para-Kählerianos en dimensión catro seguía a ser unha incógnita, a pesar dos numerosos intentos previos (ver, por exemplo, [31,32,101]). As estruturas para-Kählerianas veñen determinadas polas álxebras de Lie simplécticas que admiten unha descomposición g=L⊕L′como suma directa de dúas subálxebras Lagrangianas, polo que resultan moito máis flexibles que as estruturas Kählerianas e, en consecuencia, máis abundantes. xvi No Capítulo 3 obtemos unha descrición completa dos grupos de Lie para-Kählerianos de dimensión catro, o que nos permite describir todas as métricas para-Kählerianas localmente conformemente chás en termos de ditas estruturas no Corolario 3.2. Ademais, analizamos a curvatura de todos os grupos de Lie para-Kählerianos obtidos. No Teorema 3.1 determinamos todas as clases dos grupos para-Kählerianos simétricos. A situación non simétrica divídese no caso semi-simétrico, que se dá só cando a curvatura de Ricci se anula (ver o Teorema 3.6), e no caso non semi-simétrico, que se corresponde con catro familias entre as que se atopan os espazos 3-simétricos (ver o Teorema 3.7). Como consecuencia da nosa análise, obtivemos unha descrición alternativa dos grupos de Lie hipersimplécticos que xa foran determinados por Andrada en [5]. A existencia de métricas de Einstein na clase conforme dunha variedade dada conleva a existencia dunha estructura subxacente de C-espazo, é dicir, a existencia dun campo de vectores Xna variedade para o que se verifica a ecuación div W+ιXW= 0. Cando o campo de vectores Xé un gradiente, dita ecuación é equivalente á existencia local dunha métrica conforme gcuxo tensor de Weyl é harmónico, é dicir, tal que div W= 0, pois o tensor de Weyl de calquera variedade de Einstein ten diverxencia cero. Sobre variedades de dimensión catro orientadas, a descomposición do tensor de Weyl nas súas compoñentes auto-dual e anti-auto-dual W=W++W−fai que a condición div W= 0 resulte en div W++ div W−= 0, pois a descomposición do tensor de Weyl é conformemente invariante. Así, unha variedade de dimensión catro orientada ten tensor de Weyl medio-harmónico se, e só se, algún dos sumandos da descomposición anterior se anula. É importante salientar que esta condición non é conformemente invariante, pois div W+= div W+−ι∇σW+ para unha métrica g=e2σg. No Capítulo 6 desta memoria damos unha descrición completa dos espazos homoxéneos de dimensión catro que teñen tensor de Weyl medio-harmónico, tal e como se recolle no seguinte resultado. Teorema 6.1 Sexa (M, g)unha variedade Riemanniana localmente homoxénea de dimensión catro que ten curvatura de Weyl medio-harmónica. Entón é simétrica ou localmente homotética a unha das seguintes estensións semi-directas do grupo de Heisenberg. (i) A métrica invariante á esquerda en H3⋊ R dada por [e1, e2] = e3,[e1, e4] = −e1,[e2, e4] = −e2,[e3, e4] = −2e3, xvii (ii) ou a métrica invariante á esquerda en H3⋊ R dada por [e1, e2] = e3,[e1, e4] = 1 2e1,[e2, e4] = −e2,[e3, e4] = −1 2e3, onde {e1, e2, e3, e4}é unha base ortonormal da álxebra de Lie correspondente. É importante destacar que toda variedade localmente simétrica ten tensor de Weyl paralelo e, polo tanto, a súa diverxencia é cero. Ademais, se ben a orientación non desempeña un papel relevante no Teorema 6.1, si o fai cando se consideran estruturas adicionais sobre a variedade. Unha superficie Kähleriana satisfai div W−= 0 se, e só se, é debilmente Bochner-chá, en cuxo caso é localmente simétrica se a curvatura escalar é constante [89]. Por outra banda, como a compoñente auto-dual do tensor de Weyl dunha variedade Kähleriana de dimensión catro orientada é da forma W+=τ 12 diag[2,−1,−1], tense que τdiv1W++ι∇τW+= 0, de onde se segue que a variedade ten curvatura de Weyl medio-harmónica se a curvatura escalar é unha constante distinta de cero e, se a curvatura escalar non é constante, entón determina unha métrica conforme con tensor de Weyl medio-harmónico [56,87]. A única superficie Kähleriana homoxénea non simétrica é o espazo 3-simétrico, que necesariamente satisfai div W+= 0 e se corresponde coa métrica (ii) dada no Teorema 6.1. Os grupos de Lie no Teorema 6.1 correspóndense con variedades homoxéneas que admiten unha estrutura homoxénea auto-dual. Xeneralizando a caracterización de Cartan dos espazos simétricos como aqueles cuxo tensor de curvatura é paralelo, Ambrose e Singer [3] consideraron as estruturas homoxéneas sobre variedades Riemannianas (M, g)como campos de tensores Tde tipo (1,2) para os que se cumpre que a conexión dada por e ∇=∇−Tfai paralela á metrica g, ó tensor de curvatura Rasociado e ó campo de tensores T(que determina a torsión da conexión e ∇). Así, unha variedade Riemanniana completa e simplemente conexa é homoxénea se, e só se, admite unha estrutura homoxénea. Sekigawa probou en [126] que toda variedade Riemanniana conexa e homoxénea de dimensión tres é simétrica ou isométrica a un grupo de Lie cunha métrica invariante á esquerda (G, ⟨·,·⟩). Todo grupo de Lie é trivialmente homoxéneo considerando a acción do grupo en si mesmo dada polas translacións á esqueda e, polo tanto, posee unha estrutura homoxénea natural, denominada estrutura homoxénea canónica do grupo de Lie Riemanniano. Dado que a mesma variedade homoxénea pode admitir distintas presentacións como espazo homoxéneo, é posible que tamén admita máis dunha estrutura homoxénea. Por iso decidimos determinar todos os espazos homoxéneos Riemannianos tridimensionais que admiten máis dunha estrutura homoxénea, resultando que a situación non simétrica só ocorre cando o grupo de isometrías ten dimensión catro. xviii Teorema 7.1 Un grupo de Lie Riemanniano non simétrico e simplemente conexo de dimensión tres admite unha estrutura homoxénea distinta da estrutura canónica se, e só se, admite unha estrutura homoxénea naturalmente redutiva. Ademais, en tal caso, admite exactamente unha familia un-paramétrica de estruturas homoxéneas. Nos Teoremas 7.2 e 7.3 determinamos tamén todas as posibles estruturas homoxéneas sobre grupos de Lie Riemannianos de dimensión tres. Dado que as variedades homoxéneas de dimensión catro con tensor de Weyl medio-harmónico admiten estruturas homoxéneas auto-duais, na Sección 7.3 iniciamos o estudo destas estruturas. Se ben se coñecen resultados de clasificación baixo certas condicións adicionais (ver [109]), a descrición completa das estruturas homoxéneas auto-duais de dimensión catro parece un problema moi complexo para o que só obtivemos resultados parciais e que simplemente esbozamos na devandita sección. O tensor de Weyl de toda variedade localmente conformemente chá anúlase e, polo tanto, o funcional g7→ ZM∥W∥2dvolg acada un mínimo sobre estas variedades. En dimensión catro, este funcional é conformemente invariante e o seu gradiente vén determinado polo tensor de Bach B= div2div4W+1 2W[ρ]. Así, as variedades Bach-chás de dimensión catro son unha xeneralización natural das variedades localmente conformemente chás. O feito de que as variedades conformes Einstein, así como as (anti-)auto-duais, tamén sexan Bach-chás en dimensión catro confírelles un interese engadido. O fluxo de Bach ∂ ∂t gt=Bgt+1 12(∆τgt)gt foi intensamente estudado recentemente como un medio para mellorar o comportamento dunha métrica dada en función do seu tensor de Bach, coa idea de obter variedades Bach-chás no límite. A existencia de solitóns para o fluxo de Bach foi analizada no contexto das variedades homoxéneas produto e baixo a hipótese adicional de que o solitón fose un gradiente. No Capítulo 5 abordamos o estudo de ditos solitóns desde o punto de vista alxébrico. Un grupo de Lie Riemanniano (G, ⟨·,·⟩)é un solitón alxébrico de Bach se, e só se, D=b B−µId é unha derivación da álxebra de Lie correspondente, en cuxo caso dá lugar a un solitón de Bach ou, equivalentemente, a unha solución auto-similar gt=σ(t)ψ∗ t⟨·,·⟩, onde ψté un grupo un-paramétrico de automorfismos do grupo G. xix Dado que toda métrica de Einstein é Bach-chá, un podería esperar que os solitóns de Ricci se correspondesen con solitóns de Bach. En realidade, esta é a situación a nivel alxébrico, aínda que existen solitóns alxébricos de Bach que non son solitóns de Ricci. O seguinte resultado proporciona unha descrición completa dos solitóns alxébricos de Bach. Teorema 5.8 Un grupo de Lie Riemanniano simplemente conexo é un solitón alxébrico de Bach se, e só se, é Bach-chan, un solitón alxébrico de Ricci ou homotético a un dos seguintes grupos de Lie. (i) O grupo de Lie produto SU(2)×Rcoa métrica produto invariante á esquerda dada por [e1, e2] = 4e3,[e1, e3] = −4e2,[e2, e3] = e1. (ii) A estensión semi-directa H3⋊ R coa métrica invariante á esquerda dada por [e1, e2] = e3,[e1, e4] = ae1,[e2, e4] = 1 ae2,[e3, e4] = a2+1 ae3, para a∈(0,1). Aquí {e1, e2, e3, e4}é unha base ortonormal da álxebra de Lie correspondente. O solitón de Bach obtido a partir da afirmación (i) no teorema anterior é un solitón gradiente en S3×R, onde a métrica de S3non é a métrica redonda na esfera, senón a correspondente á dunha esfera de Berger. Pola contra, a familia (ii) non dá lugar a solitóns gradiente. Os solitóns alxébricos de Ricci foron determinados por Lauret en [98] en dimensión catro e salvo isomorfismos. No Teorema 5.4 damos unha descrición de ditos solitóns salvo homotecias, o que acurta a descrición e a fai máis manexable. As técnicas desenvolvidas no Capítulo 5 están baseadas na análise dos T-solitóns alxébricos asociados ao fluxo xeométrico xenérico ∂tgt=Tgt, onde Té un campo de tensores simétrico de tipo (0,2) sen diverxencia e invariante por isometrías. Este procedemento máis xeral levounos a simplificacións inesperadas que nos permiten traballar con fluxos xeométricos máis complexos. Os solitóns de Ricci homoxéneos en dimensións tres e catro son críticos para algún funcional cuadrático da curvatura g7→ ZM{∥ρ∥2+tτ2}dvolg con enerxía cero (ver [23]) na situación Riemanniana. Estes solitóns ou ben son gradientes ou ben son alxébricos. Os solitóns homoxéneos gradiente ou alxébricos xx Lorentzianos son tamén críticos para algún funcional cuadrático con enerxía cero en dimensión tres. Porén, e a diferencia do caso Riemanniano, a situación Lorentziana permite a existencia de solitóns de Ricci invariantes á esquerda sobre grupos de Lie. Estes solitóns foron clasificados en [24] no caso tres-dimensional. O Capítulo 4 desta memoria está adicado a abordar a clasificación dos solitóns de Ricci invariantes á esquerda en grupos de Lie Lorentzianos de dimensión catro. A situación é moito máis complexa que en dimensión tres, obtendo o seguinte resultado de clasificación. Theorem 4.2 Un grupo de Lie Lorentziano non simétrico de dimensión catro que non é unha pp-wave é un solitón de Ricci non trivial invariante á esquerda se, e só se, é homotético a un dos seguintes. (i) Gα=R3⋊ R coa álxebra de Lie dada por [e1, e4] = αe1,[e2, e4] = ε1−α2 21 2e2−e3,[e3, e4] = e2+ε1−α2 21 2e3, onde o parámetro é tal que 0≤α≤√2e{e1, e2, e3, e4}é unha base ortonormal con e3temporal. Se α= 0, entón ε= 1. Se 0< α < √2, entón ε2= 1. Neste último caso, α=2√3 3cando ε=−1. (ii) Gα=R3⋊ R coa álxebra de Lie dada por [u1, u4] = αu1,[u2, u4] = −αu2+u3,[u3, u4] = u1, α > 0, onde {u1, u2, u3, u4}é unha base pseudo-ortonormal para a que os produtos non nulos son ⟨u1, u2⟩=⟨u3, u3⟩=⟨u4, u4⟩= 1. (iii) G=E(1,1) ⋊ R coa álxebra de Lie dada por [e2, e4] = −[e1, e2] = e2,[e1, e3] = [e3, e4] = 1 2[e1, e4] = e3, onde {e1, e2, e3, e4}é unha base ortonormal con e3temporal. (iv) Gαβ =E(1,1) ⋊ R coa álxebra de Lie dada por [u1, u2] = u1,[u1, u4] = −2α(αβ + 1)u1,[u2, u3] = u3, [u2, u4] = βu1,[u3, u4] = αu3, onde {u1, u2, u3, u4}é unha base pseudo-ortonormal para a que os produtos non nulos son ⟨u1, u2⟩=⟨u3, u3⟩=⟨u4, u4⟩= 1, e os parámetros α > 0e αβ /∈−2,−1,−1 2. xxi As métricas nas afirmacións (i) e(iii) do teorema anterior teñen curvaturas de Ricci complexas, mentres que as curvaturas de Ricci das restantes afirmacións son reais. Ademais, as métricas (i),(ii) e(iv) son críticas para algún funcional cuadrático da curvatura, pero as métricas (iii) nunca o son. Estrutura da memoria No Capítulo 1 o lector pode atopar os conceptos preliminares que serán necesarios para un completo entendemento da presente memoria, que se atopa dividida en tres partes diferenciadas. A Parte I está dedicada ao estudo de estruturas localmente conformemente chás. En particular, no Capítulo 2 damos unha descrición completa das estruturas Kählerianas, para-Kählerianas e nulas-Kähler localmente conformemente chás en dimensión catro. No Capítulo 3 estudamos a realización das métricas para-Kählerianas obtidas no capítulo anterior como métricas invariantes á esquerda en grupos de Lie e damos unha descrición xeométrica completa das álxebras de Lie para-Kählerianas de dimensión catro. Na Parte II estudamos os solitóns asociados a dous fluxos xeométricos particulares: o fluxo de Ricci e o fluxo de Bach. No Capítulo 4 damos unha descrición completa dos solitóns de Ricci invariantes á esquerda en grupos de Lie Lorentzianos de dimensión catro e o Capítulo 5 está dedicado ao estudo de solitóns alxébricos en signatura Riemanniana. Nese capítulo introducimos unha técnica xeral para abordar o problema da descrición dos solitóns alxébricos Riemannianos asociados a fluxos xeométricos determinados por un tensor de tipo (0,2) simétrico e sen diverxencia xenérico e aplicámola para describir todos os solitóns alxébricos de Bach e de Ricci en dimensión catro. A Parte III cubre os Capítulos 6 e 7 desta memoria. No Capítulo 6 damos unha descrición completa das variedades homoxéneas Riemannianas que teñen curvatura de Weyl medio-harmónica. No Capítulo 7 determinamos todos os espazos homoxéneos Riemannianos que admiten máis dunha estrutura homoxénea. Ademais, motivados polo feito de que as métricas obtidas no Capítulo 6 admiten estruturas homoxéneas auto-duais, comezamos a estudar estas estruturas en dimensión catro e pódese atopar un bosquexo do problema na Sección 7.3. Este segue sendo un problema aberto para o que, ata o de agora, só obtivemos resultados parciais. xxii Objectives and Methodology Objectives The main objectives of this thesis are the following. The reader can find a motivation and a detailed description of them in the introduction of this thesis. O.1 To completely understand the local structure of the four-dimensional Kähler and para-Kähler metrics that have at least one flat representative in their conformal classes. O.2 To complete the description of left-invariant para-Kähler structures on fourdimensional Lie groups. O.3 The complete description of all the four-dimensional Riemannian homogeneous spaces that have half-harmonic Weyl curvature. O.4 To determine all the three-dimensional Riemannian homogeneous spaces that admit more than one homogeneous structure. O.5 The investigation of four-dimensional Riemannian self-dual homogeneous structures. O.6 The complete description of four-dimensional Riemannian algebraic Bach solitons. O.7 The complete description of four-dimensional left-invariant Lorentzian Ricci solitons. Methodology I include this section here since it is mandatory to do so in the University of Santiago de Compostela. Every step I took throughout this thesis is clearly detailed in the corresponding chapter, but in the following lines I will outline the main aspects of the general methodology followed in mathematical research. xxiii Such methodology consists of the study of behavioural patterns of mathematical structures and their subsequent logical analysis, which allows us to find the general properties of the mathematical objects that constitute the subjects of our studies. In order to carry out this process, one first needs to acquire the most basic knowledge required for the understanding of the more complicated theory we intend to build upon it. Therefore, the first step is to read research papers regarding previous results on the matter of interest. One may find unmanageable calculations on the way, and it is sometimes necessary to use computing methods and symbolic calculus tools to tackle them. This requires to learn some programming techniques in order to simplify the calculations. Finally, it is important to keep in touch with experts on the fields of differential geometry and geometric analysis. Besides the frequent meetings with my Ph.D. advisors, the discussions kept during the conferences I attended in the past years were essential for the development of the results covered by the present thesis. xxiv non-symmetric situation splits into the semi-symmetric case – that occurs only when the corresponding Ricci tensors vanish – and the non-semi-symmetric case – which corresponds to four families among which we found some of the 3-symmetric spaces – as stated in Theorems 3.6 and 3.7. As a consequence of our study, we obtained an alternative description of the hypersymplectic Lie groups previously obtained by Andrada [5]. The existence of Einstein metrics in the conformal class of a given metric implies the existence of an underlying C-space structure, i.e., the existence of a vector field Xon the manifold such that div W+ιXW= 0. When the vector field Xis the gradient of a function, this equation is equivalent to the existence of a conformal metric gfor which div W= 0, since the Weyl curvature of an Einstein manifold is harmonic – i.e., divergence-free. On four-dimensional oriented manifolds, the decomposition of the Weyl curvature tensor into its self-dual and anti-self-dual components W=W++W−, which is conformally invariant, makes the condition div W= 0 become div W++ div W−= 0. In this way, a four-dimensional oriented manifold has half-harmonic Weyl curvature if either one of the two summands in the previous equation vanishes. It is important to emphasize that this condition is not conformally invariant, since div W+= div W+−ι∇σW+ for a conformal metric g=e2σg. Four-dimensional homogeneous Riemannian manifolds have harmonic Weyl curvature tensor if and only if they are symmetric [123]. In Chapter 6 we give the complete description of the four-dimensional homogeneous spaces that have half-harmonic Weyl curvature, as stated in the main result of the chapter. Theorem 6.1 Let (M, g)be a four-dimensional locally homogeneous Riemannian manifold with half-harmonic Weyl curvature tensor. Then it is symmetric or locally homothetic to one of the following semi-direct extensions of the Heisenberg group. (i) The left-invariant metric on H3⋊ R determined by [e1, e2] = e3,[e1, e4] = −e1,[e2, e4] = −e2,[e3, e4] = −2e3, (ii) or the left-invariant metric on H3⋊ R determined by [e1, e2] = e3,[e1, e4] = 1 2e1,[e2, e4] = −e2,[e3, e4] = −1 2e3, xxxi Introduction where {e1, e2, e3, e4}is an orthonormal basis of h3⋊r. It is important to emphasize that the Weyl curvature tensor of a locally symmetric manifold is parallel and so it is divergence-free. Besides, even though orientation does not play a relevant role in Theorem 6.1, it does when additional structures on the manifold are considered. A Kähler surface satisfies div W−= 0 if and only if it is weakly Bochner-flat, in which case it is locally symmetric when its scalar curvature is constant (see [89]). Furthermore, since the self-dual Weyl curvature of an oriented four-dimensional Kähler metric takes the form W+=τ 12 diag[2,−1,−1], then τdiv1W++ι∇τW+= 0 and so div1W+= 0 if the scalar curvature is a non-zero constant. If the scalar curvature is not constant, then it determines a conformal metric whose Weyl curvature tensor is half-harmonic [56, 87]. The only non-symmetric homogeneous Kähler metric is the 3-symmetric space, which is necessarily half-harmonic, and it corresponds to Theorem 6.1-(ii). The Lie groups given in Theorem 6.1 correspond to homogeneous manifolds that admit self-dual homogeneous structures. Generalizing Cartan’s characterization of symmetric spaces as those whose curvature tensors are parallel, Ambrose and Singer [3] considered homogeneous structures on a Riemannian manifold (M, g)as tensor fields of type (1,2) for which the connection e ∇=∇ − Tmakes the metric g, its associated curvature tensor Rand the tensor field Tparallel. In this way, a complete and simply connected Riemannian manifold is homogeneous if it admits a homogeneous structure. Sekigawa proved in [126] that any three-dimensional simply connected homogeneous Riemannian manifold is either symmetric or isometric to a Lie group endowed with a left-invariant metric (G, ⟨·,·⟩). Lie groups are trivially homogeneous, just considering the action of Gon itself determined by the left translations. Therefore, all of them have a natural homogeneous structure, called the canonical homogeneous structure of the Riemannian Lie group. Given the fact that a homogeneous manifold may admit different presentations as a homogeneous space, it may as well admit more than one homogeneous structure. Therefore, we decided to try and determine all the three-dimensional Riemannian homogeneous spaces that admit more than one homogeneous structure, and saw that in the non-symmetric situation this occurs only when their isometry groups are fourdimensional. Theorem 7.1 A non-symmetric simply connected three-dimensional Riemannian Lie group admits a homogeneous structure different from the canonical one if and only if it admits a naturally reductive homogeneous structure. Moreover, in such a case, it admits exactly a one-parameter family of homogeneous structures. xxxii In Theorems 7.2 and 7.3 we also determine all the possible homogeneous structures on three-dimensional Riemannian Lie groups. Since the four-dimensional homogeneous manifolds that have half-harmonic Weyl curvature admit self-dual homogeneous structures, we started the study of such structures in Section 7.3. Even though there already are classification results under certain additional conditions [109], the complete description of four-dimensional self-dual homogeneous structures seems to be an arduous problem that we only sketch in the mentioned section. The Weyl curvature tensor of any locally conformally flat manifold vanishes and so the functional g7→ ZM∥Wg∥2dvolg reaches a minimum on every such manifold. This functional is conformally invariant in the four-dimensional situation and its gradient is determined by the Bach tensor B= div2div4W+1 2W[ρ]. Therefore, Bach-flat four-dimensional manifolds are a natural generalization of locally conformally flat manifolds. The fact that four-dimensional both conformally Einstein and (anti-)self-dual manifolds are also Bach-flat places more importance on the study of such manifolds. The Bach flow ∂ ∂t gt=Bgt+1 12(∆τgt)gt has been intensively studied recently with the intention of improving the behaviour of a given metric in terms of its Bach tensor and obtaining Bach-flat metrics at the limit. The existence of solitons associated to this flow has been studied in the context of product homogeneous manifolds and under the additional hypothesis of the soliton being a gradient. In Chapter 5 we tackle the study of such solitons from the algebraic point of view. A Riemannian Lie group (G, ⟨·,·⟩)is an algebraic Bach soliton if and only if D=b B−µId is a derivation of the Lie algebra of G, in which case it gives rise to a Bach soliton or, equivalently, to a self-similar solution gt=σ(t)ψ∗ t⟨·,·⟩, where ψtis one-parameter group of automorphisms of G. Since every Einstein metric is Bach-flat, one might expect Ricci solitons to correspond to Bach solitons. This is actually the situation at the algebraic level, although there are algebraic Bach solitons which are not Ricci solitons. The following result provides a complete description of four-dimensional Riemannian algebraic Bach solitons. xxxiii Introduction Theorem 5.8 A four-dimensional simply connected Riemannian Lie group is an algebraic Bach soliton if and only if it is Bach-flat, an algebraic Ricci soliton or homothetic to one of the following Lie groups. (i) The product Lie group SU(2) ×Rwith the product left-invariant metric determined by [e1, e2] = 4e3,[e1, e3] = −4e2,[e2, e3] = e1. (ii) The semi-direct extension H3⋊ R with the left-invariant metric determined by [e1, e2] = e3,[e1, e4] = ae1,[e2, e4] = 1 ae2,[e3, e4] = a2+1 ae3, for a∈(0,1). Here {e1, e2, e3, e4}denotes an orthonormal basis of the corresponding Lie algebra. The Bach soliton obtained from Assertion (i) is a gradient soliton on S3×R, where the metric on S3is not the round metric, but a Berger one. On the other hand, the family of metrics in Assertion (ii) does not give rise to gradient solitons. The four-dimensional algebraic Ricci solitons were determined – up to isomorphisms – by Lauret in [98]. In Theorem 5.4 we give a description of such solitons up to homotheties, which makes the description shorter and more manageable. The techniques developed in Chapter 5 are based on the analysis of general algebraic T-solitons for a geometric flow ∂tgt=Tgtgiven by an isometrically invariant symmetric, divergence-free (0,2)-tensor field. This general approach led to unexpected simplifications which enable us to consider more complicated geometric flows. Homogeneous Ricci solitons are critical for some curvature quadratic functional g7→ ZM∥ρ∥2+tτ2dvolg with zero energy in dimensions three and four (see [23]) and Riemannian signature. These solitons are algebraic or gradient solitons. In the same way, homogeneous gradient or algebraic Lorentzian Ricci solitons are critical for some curvature quadratic functional with zero energy in dimension three. Nevertheless, and in sharp contrast with the Riemannian situation, the Lorentzian signature allows the existence of leftinvariant Ricci solitons on Lie groups. Such solitons had already been classified in the three-dimensional situation [24], and we tackle the four-dimensional problem in Chapter 4 of this thesis. The situation is far more complicated than the threedimensional one and we obtained the following result. Theorem 4.2 A non-symmetric four-dimensional Lorentzian Lie group which is not a pp-wave is a non-trivial left-invariant Ricci soliton if and only if it is homothetic to one of the following: xxxiv (i) Gα=R3⋊ R with Lie algebra given by [e1, e4] = αe1,[e2, e4] = ε1−α2 21 2e2−e3,[e3, e4] = e2+ε1−α2 21 2e3, where the parameter satisfies 0≤α≤√2and {e1, e2, e3, e4}is an orthonormal basis with timelike e3. If α= 0, then ε= 1. If 0< α < √2, then ε2= 1. In the latter case, α=2√3 3when ε=−1. (ii) Gα=R3⋊ R with Lie algebra given by [u1, u4] = αu1,[u2, u4] = −αu2+u3,[u3, u4] = u1, α > 0, where {u1, u2, u3, u4}is a pseudo-orthonormal basis for which the non-zero inner products are ⟨u1, u2⟩=⟨u3, u3⟩=⟨u4, u4⟩= 1. (iii) G=E(1,1) ⋊ R with Lie algebra given by [e2, e4] = −[e1, e2] = e2,[e1, e3] = [e3, e4] = 1 2[e1, e4] = e3, where {e1, e2, e3, e4}is an orthonormal basis with timelike e3. (iv) Gαβ =E(1,1) ⋊ R with Lie algebra given by [u1, u2] = u1,[u1, u4] = −2α(αβ + 1)u1,[u2, u3] = u3, [u2, u4] = βu1,[u3, u4] = αu3, where {u1, u2, u3, u4}is a pseudo-orthonormal basis for which the non-zero inner products are ⟨u1, u2⟩=⟨u3, u3⟩=⟨u4, u4⟩= 1, and the parameters α > 0and αβ /∈−2,−1,−1 2. The metrics in Assertions (i) and (iii) have complex Ricci curvatures, while those of the remaining two are real. Besides, the metrics in Assertions (i),(ii) and (iv) are critical for some curvature quadratic functionals, whereas the metrics in Theorem 4.2- (iii) never are. The case of left-invariant Ricci solitons which are pp-wave and plane wave Lie groups are described in Theorems 4.9 and 4.11, respectively. xxxv Introduction The outline of this thesis In Chapter 1 the reader can find the preliminary concepts that will be necessary for the complete understanding of the contents of this thesis, which is divided in three distinguished parts. Part I is devoted to the study of locally conformally flat structures. In particular, in Chapter 2 we give a complete description of four-dimensional locally conformally flat Kähler, para-Kähler and null-Kähler structures. In Chapter 3 we study the realization of the para-Kähler families obtained in the previous chapter as left-invariant metrics on Lie groups and give a complete geometric description of the four-dimensional para-Kähler Lie algebras. In Part II we study the solitons associated to two particular geometric flows: the Bach flow and the Ricci flow. In Chapter 4 we give a complete description of the four-dimensional left-invariant Lorentzian Ricci solitons and Chapter 5 is devoted to the study of the algebraic situation in Riemannian signature. In this chapter we introduce a general technique to describe four-dimensional Riemannian algebraic solitons associated to the flow determined by a generic symmetric and divergence-free (0,2)-tensor field and use it to describe all the algebraic Bach and Ricci solitons in dimension four. Part III covers Chapters 6 and 7 of this thesis. In Chapter 6 we give a complete description of the four-dimensional homogeneous Riemannian manifolds that have half-harmonic Weyl curvature. In Chapter 7 we determine all the non-symmetric three-dimensional Riemannian homogeneous spaces that admit more than one homogeneous structure. Besides, motivated by the fact that the metrics obtained in Chapter 6 admit self-dual homogeneous structures, we began to study these sort of structures in dimension four and the reader can find a sketch of this problem in Section 7.3. This remains an open problem for which we have obtained only partial results so far. xxxvi Chapter 1 Preliminaries In this chapter we will introduce the concepts and notations that will be necessary for the complete understanding of this thesis. We will omit most of the proofs and remit the reader to different bibliographic references for more details. 1.1 Pseudo-Riemannian geometry 1.1.1 Pseudo-Riemannian manifolds An n-dimensional pseudo-Riemannian manifold is a pair (M, g)where Mis a smooth manifold and gis a metric tensor, i.e., a symmetric and non-degenerate (0,2)-tensor field on M. The signature of the metric gis the pair (n−ν, ν)such that n−νand ν are the number of negative and positive eigenvalues of its associated matrix, respectively. Let (M, g)be an n-dimensional pseudo-Riemannian manifold. It is said to be Riemannian if its signature is (0, n)and Lorentzian if its signature is (1, n−1). Moreover, if Mis even-dimensional and the signature of gis n 2,n 2then the manifold has neutral (or split) signature. We will denote by T M and T∗Mthe tangent and cotangent bundles of the manifold Mand by X(M)the space of vector fields which are tangent to M. We will use capital letters to denote vector fields and small letters to denote tangent vectors at a given point. Given a non-zero vector v∈TpMtangent to Mat a point p∈M, it is said to be timelike if g(v, v)<0,spacelike if g(v, v)>0and null or lightlike if g(v, v) = 0. For any pseudo-Riemannian manifold (M, g)there exists a unique adapted linear connection ∇which is torsion-free and parallel, i.e., such that ∇XY−∇YX−[X, Y ] = 0 and ∇g= 0. This connection is known as the Levi-Civita connection of the pseudo-Riemannian manifold and it is characterized by the Koszul formula 2g(∇XY, Z) = Xg (Y, Z) + Y g (X, Z)−Zg (X, Y ) −g(X, [Y, Z]) −g(Y, [X, Z]) + g(Z, [X, Y ]) , 1 Preliminaries where X, Y, Z ∈X(M)and [·,·]denotes the Lie bracket. The Levi-Civita connection can also be described by means of the Christoffel symbols. Let x1, . . . , xnbe local coordinates on M. The Christoffel symbols of the first kind are given by Γijℓ =1 2∂gℓj ∂xi+∂gℓi ∂xj−∂gij ∂xℓ and so the Christoffel symbols of the second kind are Γijk=gkℓΓijℓ, being gijthe inverse matrix of (gij). Therefore the Levi-Civita connection can be written in coordinates as ∇∂xi∂xj= Γijk∂xk, where ∂xi:= ∂ ∂xidenote the locally defined coordinate vector fields. 1.1.2 The curvature tensor In terms of the Levi-Civita connection we can define the curvature operator, or curvature tensor of type (1,3) by the convention R(X, Y )Z=∇[X,Y ]Z−[∇X,∇Y]Z. Considering local coordinates x1, . . . , xnon M, the components of the curvature operator are given by R(∂xi, ∂xj)∂xk=Rijkℓ∂xℓ. We can obtain the curvature tensor of type (0,4) by lowering indices in the previous expression R(X, Y, Z, V ) = g(R(X, Y )Z, V ), so that its components are given by Rijkℓ =gℓrRijkr. Moreover, the curvature tensor has the following algebraic properties (i)R(X, Y, Z, V ) = −R(Y, X, Z, V ) = −R(X, Y, V, Z), (ii)R(X, Y, Z, V ) + R(Y, Z, X, V ) + R(Z, X, Y, V ) = 0, (iii)R(X, Y, Z, V ) = R(Z, V, X, Y ), (1.1) and the differential identity (iv) (∇XR) (Y, Z, U, V )+(∇YR) (Z, X, U, V )+(∇ZR) (X, Y, U, V ) = 0. Identities (ii)and (iv)are known as the first and second Bianchi identities, respectively. A (0,4)-tensor field A:V ×V ×V ×V → Ron a vector space Vsatisfying identities (1.1) is said to be an algebraic curvature tensor. 2 1.1 Pseudo-Riemannian geometry The sectional curvature of a given Riemannian manifold (M, g)is the real function Kdefine on the Grassmannian of 2-planes by K(Π) = R(X, Y, X, Y ) g(X, X)g(Y, Y )−g(X, Y )2, where Π = span{X, Y }is a two-dimensional subspace of TpM. In the pseudo-Riemannian case we must consider the restriction to the Grassmannian of non-degenerate planes, i.e., those planes such that g(X, X)g(Y, Y )−g(X, Y )2= 0. If K(Π) is independent of the plane Π, then the curvature tensor is given by R(X, Y, Z, V ) = KR0(X, Y, Z, V ), where R0is the standard algebraic curvature tensor given by R0(X, Y, Z, V ) = g(X, Z)g(Y, V )−g(X, V )g(Y, Z).(1.2) In dimension greater than two, if Mis connected then the second Bianchi identity implies that if Kis pointwise constant, then it is necessarily a global constant. A pseudo-Riemannian manifold has constant sectional curvature Kif and only if its curvature tensor can be written as R(X, Y )Z=K{g(X, Z)Y−g(Y, Z)X}, in which case the manifold is locally isometric to a pseudo-sphere Sn ν(when K > 0), to a pseudo-Euclidean space En ν(when K= 0) or to a pseudo-hyperbolic space Hn ν (when K < 0). We refer to O’Neil’s book [112] for more details on this topic. We will denote by ρthe Ricci tensor, which is defined as the second trace of the curvature operator, ρ(X, Y ) = tr (Z7→ R(X, Z)Y) and its associated (1,1)-tensor field, known as the Ricci operator, is characterized by g(Ric(X), Y ) = ρ(X, Y ). The scalar curvature is defined as the trace of the Ricci operator τ= tr(Ric). It follows from the curvature identities (1.1) that the Ricci tensor is symmetric and, equivalently, the Ricci operator is self-adjoint. The Ricci tensor and the scalar curvature can be expressed in coordinates as ρij =grℓRirjℓ, τ =gijρij. Any two-dimensional pseudo-Riemannian manifold satisfies ρ=τ 2g. A pseudoRiemannian manifold of dimension greater than two is said to be an Einstein space if 3 Preliminaries its Ricci tensor is a constant multiple of the metric, ρ=λg. Tracing on the previous expression one sees that ρ=τ ng, (1.3) and if Mis connected, then the second Bianchi identity leads to the constancy of τ. In dimension three, satisfying the Einstein condition (1.3) is equivalent to having constant sectional curvature, while in dimension four there exist Einstein metrics which are not of constant sectional curvature. The four-dimensional case then appears as the first non-trivial case for consideration. 1.1.3 The Weyl tensor The Schouten tensor of an algebraic curvature tensor Aon an n-dimensional inner product vector space (V,⟨·,·⟩)is the symmetric (0,2)-tensor field defined as SA=1 n−2ρA−τA 2(n−1)⟨·,·⟩, where ρAand τAare the Ricci tensor and the scalar curvature associated to the algebraic curvature tensor A. Let Dand Bbe two symmetric bilinear forms on a vector space V. Their Kulkarni-Nomizu product D⊙Bis the (0,4)-tensor field on Vdefined as (D⊙B) (x, y, z, v) = D(x, z)B(y, v) + D(y, v)B(x, z) −D(x, v)B(y, z)−D(y, z)B(x, v), for x, y, z, v ∈ V. It is easy to check that D⊙Bis an algebraic curvature tensor on (V,⟨·,·⟩). For example, the standard curvature tensor R0=1 2⟨·,·⟩⊙⟨·,·⟩. The Weyl curvature tensor arises from the Kulkarni-Nomizu product of the Schouten tensor and the metric tensor as WA=A−SA⊙ ⟨·,·⟩. Therefore, the Weyl curvature tensor of a pseudo-Riemannian manifold (M, g)is defined as W=R−S⊙g, which can be written at each point p∈Mas W(x, y, z, v) = R(x, y, z, v) + τ (n−1)(n−2) {g(x, z)g(y, v)−g(x, v)g(y, z)} −1 n−2{ρ(x, z)g(y, v)−ρ(x, v)g(x, z) + ρ(y, v)g(x, z)−ρ(y, z)g(x, v)}, for all x, y, z, v ∈TpM. An important property of the Weyl curvature tensor is that it is trace-free and in dimension three it vanishes identically. 4 1.2 Curvature functionals ∇T =−∆ρ+∇2τ−1 2∆τg −2R[ρ] + 1 2∥ρ∥2g, ∇R =−4∆ρ+ 2∇2τ−2ˇ R+1 2∥R∥2g−4R[ρ] + 4ˇρ. The curvature tensor Rof a three-dimensional pseudo-Riemannian manifold is completely determined by its Ricci tensor, which means that the three curvature scalar invariants satisfy the identity ∥R∥2= 2∥ρ∥2−1 2τ2. Consequently, the functional determined by the L2-norm of the curvature tensor can be expressed as a linear combination of the other two as R= 2T − 1 2S in the three-dimensional case. In the four-dimensional setting, the Chern-GaussBonnet Theorem gives the Euler characteristic of a compact pseudo-Riemannian manifold with no boundary in terms of the three curvature scalar invariants as χ(M) = 1 8π2ZM∥R∥2−4∥ρ∥2+τ2dvolg. Therefore, the four-dimensional curvature functional Ris determined by R=tπ2χ(M)+4T −S, and, since the Euler characteristic of a manifold is a topological invariant, the critical points of Rcorrespond to the critical points of 4T − 1 4S. Therefore, the functionals Rand T −1 4Sare equivalent. This shows that it is enough to study the functionals S:g7→ ZM τ2dvolg,Ft:g7→ ZM∥ρ∥2+tτ2dvolg, for all t∈R. The gradients of the functionals Ftare given by ∇Ft=−∆ρ+ (1 + 2t)∇2τ−1+4t 2∆τg −2R[ρ] + 1 2∥ρ∥2g+1 2tτ2g−2tτρ. Proceeding in the same way as we did for the Hilbert-Einstein functional, when we restrict our study to the space of metrics of constant volume, we obtain that the EulerLagrange equations corresponding to these functionals are given by ∇2τ−1 n∆τg −τρ−τ ng= 0, ∆ρ−(1 + 2t)∇2τ+2 nt∆τg + 2 R[ρ]−1 n∥ρ∥2g+ 2tτ ρ−τ ng= 0. Note that Einstein metrics are critical for the functionals Sand Ftfor any value of t. Consequently, Einstein metrics are critical for all the curvature quadratic functionals in dimensions three and four, but this is no longer true in higher dimensions. 11 Preliminaries The L2-norm of the Weyl curvature tensor: the Bach tensor In dimension four, the functional given by W:g7→ W(g) = ZM∥Wg∥2dvolg, where Wgdenotes the Weyl curvature tensor associated to g, quantifies the deflection of a Riemannian metric gfrom being locally conformally flat. A remarkable property of this functional is that it is conformally invariant in dimension four. Indeed, if ¯g=e2σg, then ∥W∥2dvol¯g=WijkℓWijkℓdvol¯g =e2σWijkℓe2σe−8σWijkℓe4σdvolg=∥W∥2dvolg. Furthermore, it follows from the Chern-Gauss-Bonnet Theorem that W(g) = 32π2χ(M)+2F−1/3, and so this functional is equivalent to F−1/3. The conformal invariance of Wshows that four-dimensional conformally Einstein metrics are F−1/3-critical. Such metrics are equivalently characterized by the vanishing of their Bach tensor (see [10]), which is defined as the (0,2)-tensor field given by B= div2div4W+1 2W[ρ], where W[ρ]ij =Wijkℓρkℓ. In addition to conformally Einstein metrics, half conformally flat metrics are also Bach-flat in dimension four. Recall that the Hirzebruch signature formula allows us to express the Hirzebruch signature as τ[M] = 1 12π2ZM∥W+∥2−∥W−∥2dvolg, and so the functional Wcan be written as W(g) = RM∥W∥2dvolg=RM∥W+∥2+∥W−∥2dvolg =±12π2τ[M]+2RM∥W∓∥2dvolg, which shows that half conformally flat metrics are critical for the functional W, and therefore Bach-flat. Besides, since the functional Wis equivalent to the one given by the L2-norm of the self-dual Weyl curvature tensor W+, if gis a Kähler metric (see 12 1.3 Affine and projective geometry Part I), whose self-dual conformal operator W+is diagonalizable and takes the form W+=τ 12 diag[2,−1,−1], then W(g) = −12π2τ[M]+2ZM∥W+∥2dvolg=−12π2τ[M] + 1 12 ZM τ2dvolg. Consequently, the Calabi functional – which is the restriction of Sto metrics in the same Kähler class – is equivalent to F−1/3when restricted to such variations (see [29,127]). 1.3 Affine and projective geometry An affine manifold is a pair (M, D)where Mis a smooth manifold and Dis an affine torsion-free connection on M. The Ricci tensor associated to Dis defined as Dρ(X, Y ) := tr Z→DR(X, Z)Y, and, since it need not be symmetric in general, we introduce the symmetrization Dρs and skew-symmetrization Dρsk as Dρs(X, Y ) := 1 2Dρ(X, Y ) + Dρ(Y, X), Dρsk(X, Y ) := 1 2Dρ(X, Y )−Dρ(Y, X).(1.6) An affine manifold is flat if its associated curvature tensor DRvanishes. In this case, there exist local coordinates where the Christoffel symbols are zero. Aprojective structure on an affine manifold (M, D)is an equivalence class [D] of affine connections on TM sharing the same unparametrized geodesics. Two affine connections Dand e Dare projectively equivalent if and only if there exists a one-form ω=ωidxion Msuch that e Γijk= Γijk+δikωj+δjkωi, where Γijkand e Γijkdenote the Christoffel symbols of the affine connections Dand e D, respectively, and δikdenotes the Kronecker delta. According to this, an affine manifold (M, D)is said to be projectively flat if there exists a flat representative in the class [D], i.e., if there exists a one-form ωon Msuch that Γijk=−δikωj+δjkωi.(1.7) Furthermore, if there exists a real-valued function flocally defined on Msuch that ω=df, then (M, D)is said to be locally strongly projectively flat. Two-dimensional projectively flat affine manifolds are characterized as follows (see [110]). 13 Preliminaries Theorem 1.2. Let (Σ, D)be an affine surface. Then (Σ, D)is projectively flat if and only if Dρand DDρare totally symmetric. An affine manifold is said to be curvature recurrent (respectively, Ricci recurrent) if DDR=ξ⊗DR(respectively, DDρ=ξ⊗Dρ) for some one-form ξon Mand (M, D)is locally symmetric if DDR= 0. Since the curvature tensor of any affine surface is determined by the associated Ricci tensor as DR(X, Y )Z=Dρ(X, Z)Y−Dρ(Y, Z)X, curvature recurrence and Ricci recurrence are equivalent in the two-dimensional case. 1.3.1 Walker structures Let (M, g)be a pseudo-Riemannian manifold and consider a distribution Dof the tangent space. Dis parallel if ∇XY∈ D for every smooth vector field Xand every Y∈ D, and degenerate if g|Dvanishes. It is well-known that the existence of a parallel distribution on a Riemannian manifold induces a local de Rham decomposition as a product. This property is also true in the pseudo-Riemannian setting as long as the parallel distribution is nondegenerate. The situation in which the distribution is degenerate was studied by Walker [132], who gave a canonical expression for this kind of metrics. This is why pseudo-Riemannian manifolds admitting a parallel degenerate distribution are called Walker manifolds. Walker metrics give rise to many strictly pseudo-Riemannian situations, such as degenerate homogeneous pseudo-Riemannian structures, strictly conformally symmetric manifolds, conformally flat metrics with two-step nilpotent Ricci operator, Einstein hypersurfaces in manifolds with constant sectional curvature and nilpotent shape operator, para-Kähler structures and so on. In the most general situation, the existence of adapted Walker coordinates is given by the following result (see [132]). Theorem 1.3. Let (M, g)be an n-dimensional Walker manifold and let Dbe an rdimensional, parallel degenerate distribution. There exists a local system of adapted coordinates x1, . . . , xn−r, xn−r+1, . . . , xnon Mwith respect to which the metric tensor takes the form (gij) =   B H Idr t H A 0 Idr0 0  , where Idris the identity matrix of order rand A,Band Hare matrices whose components are functions of the adapted coordinates satisfying the following conditions: 14 1.3 Affine and projective geometry 1. Ais an (n−2r)×(n−2r)matrix and Bis an r×rmatrix. 2. His an r×(n−2r)matrix and t Hdenotes its transpose matrix. 3. Aand Hare independent of the last n−rcoordinates. Moreover, the distribution Dis spanned by the last n−rcoordinate vector fields. The canonical form given in the previous theorem turns out to be simpler when the manifold has even dimension n= 2mand the Walker distribution is of maximal dimension. In this case, there exist local Walker coordinates x1, . . . , xm, x1′, . . . , xm′ in which the metric tensor takes the form g=BIdm Idm0,(1.8) where Bis a matrix whose components are functions of x1, . . . , xm, x1′, . . . , xm′. This kind of metrics will play an important role in the work developed in this thesis. Their non-zero Christoffel symbols and curvature operator are given by the following results (see [41]). Lemma 1.4. Let (M, g, D)be a Walker manifold of dimension n= 2m, where dim D=m. Then its Christoffel symbols are given, up to the corresponding symmetries, by Γk ij =−1 2∂k′gij, Γk′ i′j=1 2∂i′gjk, Γk′ ij =1 2(−∂kgij +∂jgik +∂igjk +gks∂s′gij), where 1≤s≤m. Lemma 1.5. Let (M, g)be a Walker manifold of dimension n= 2mwith mdimensional parallel degenerate distribution D. The non-zero components of its curvature tensor of type (1,3) are given, up to the corresponding symmetries, by Rh ijk =−1 2(∂j∂h′gik −∂i∂h′gjk)−1 4(∂s′gjk∂h′gis −∂s′gik∂h′gjs), Rh′ ijk =−1 2(∂i∂kgjh −∂i∂hgjk +∂j∂hgik −∂j∂kgih) −1 4{∂s′gjk (∂hgis −∂sgih −∂igsh −ght∂t′gis) −∂s′gik (∂hgjs −∂sgjh −∂jgsh −ght∂t′gjs) −∂s′gih (∂sgjk −∂kgjs −∂jgks −gst∂t′gjk) +∂s′gjh (∂sgik −∂kgis −∂igks −gst∂t′gik) +2∂i(ghs∂s′gjk)−2∂j(ghs∂s′gik)}, 15 Preliminaries Rh ij′k=−1 2∂j′∂h′gik, Rh′ ij′k=−1 2∂h∂j′gik −∂k∂j′gih −1 4∂s′gik∂j′gsh +∂s′gih∂j′gsk −2∂j′(ghs∂s′gik), Rh′ ijk′=−1 2(∂i∂k′gjh −∂j∂k′gih)−1 4(∂k′gjs∂s′gih −∂k′gis∂s′gjh), Rh′ ij′k′=1 2∂j′∂k′gih, where 1≤s≤mand 1≤t≤m. In general, the fact that Dis parallel implies that the curvature tensor of any Walker manifold is such that R(D,D⊥,·,·)=0, R(D,D,·,·)=0,and R(D⊥,D⊥,D,·)=0. 1.3.2 Metrics on the cotangent bundle A particular class of Walker metrics are those known as Riemannian extensions. The interest of these metrics resides in the fact that they allow the translation of problems in affine geometry to problems in pseudo-Riemannian geometry and vice versa. Let T∗Mdenote the cotangent bundle of an n-dimensional smooth manifold and π:T∗M→Mbe the natural projection from the cotangent bundle onto the base manifold. A point ˜p∈T∗Mis of the form ˜p= (p, ω), where p:= π(˜p)∈Mand ω∈T∗ pM. We will give some basic notions about the geometry of the cotangent bundle before introducing Riemannian extensions. Let ˜p= (p, ω)be a point in T∗Mand consider local coordinates x1, . . . , xn on a neighbourhood Uof the point p∈M. In this neighbourhood we can write ω=xi′dxi, which allows us to define a system of local coordinates on e U:= π−1(U)⊂T∗Mas x1, . . . , xn, x1′, . . . , xn′. In terms of these local coordinates, the canonical symplectic structure of the cotangent bundle is given by Ω := dω =dxi′∧dxi.(1.9) Let us consider a vector field Xon M. Its evaluation map ιX is the differentiable map on the cotangent bundle given by ιX(p, ω) = ω(Xp). 16 1.3 Affine and projective geometry If we write X=Xi∂xi, where Xi=dxi(X), then ιX xi, xi′=xi′Xi. Vectors fields on T∗Mare determined by their action on evaluation maps of vector fields on M. In this way, two vector fields e Yand e Zon Mare the same if and only if e Y(ιX) = e Z(ιX)for every vector field Xon M. Bearing this in mind, the complete lift of a vector field Xon Mis a vector field on T∗Mcharacterized by the identity XC(ιZ) = ι[X, Z], for every vector field Zon M. The tangent space to the cotangent bundle of a smooth manifold is generated by the complete lifts of all smooth vector fields on M. Complete lifts allow us to characterize tensor fields of type (0, s)in the sense that two tensor fields ˜ Sand ˜ Tare the same if and only if e TXC 1, . . . , XC s=e SXC 1, . . . , XC s for any vector fields X1, . . . , Xson M. Knowing this, it is easy to see that the twoform (1.9) does not depend on the system of local coordinates in consideration and is characterized by the identity ωXC, Y C=ι[X, Y ]. Now, if Tis a (1,1)-tensor field on M, it is an endomorphism of the tangent bundle TM, so one can define a one-form ιT on the cotangent bundle characterized by the identity ιT XC=ι(TX), which takes the form ιT =xk′Tikdxiwith respect to the coordinates induced on T∗M. Riemannian extensions The construction of this kind of metrics defines a Walker metric on the cotangent bundle of an affine n-manifold (M, D), where Ddenotes a torsion-free connection on M.T∗Mcan be equipped with a pseudo-Riemannian metric of neutral signature (n, n)given by the identity gDXC, Y C=−ι(DXY+DYX). This metric is called a Riemannian extension (see [120]) and in terms of the system of local coordinates induced on T∗Mtakes the form gD=−2xk′ΓijkIdn Idn0, i, j = 1, . . . , n, i′=i+n, (1.10) 17 Preliminaries where Γijkare the Christoffel symbols of the affine connection Dwith respect to the coordinates xion M. These metrics are particular cases of Walker metrics for which the parallel, degenerate distributions have maximal dimension and are given by D= ker π∗. As we have already mentioned, Riemannian extensions provide a link between affine geometry and Riemannian geometry so that some properties of the affine connection D can be studied through the properties of gD. For instance, Dis projectively flat if and only if gDis locally conformally flat. Deformed Riemannian extensions These metrics are a slight generalization of Riemannian extensions involving an additional (0,2)-tensor field on Mthat we will call Φ. Deformed Riemannian extensions are again pseudo-Riemannian metrics of neutral signature (n, n)on the cotangent bundle of an affine manifold and are given by gD,ΦXC, Y C=gD+π∗Φ =−ι(DXY+DYX) + π∗Φ. Deformed Riemannian extensions can be expressed in terms of the coordinates induced on the cotangent bundle as gD,Φ=−2xk′Γijk+ Φij Idn Idn0, i, j = 1, . . . , n, i′=i+n. A criterium to characterize deformed Riemannian extensions amongst Walker manifolds was shown in [1], where it is stated that if (M, g, D)is a Walker manifold then gis a deformed Riemannian extension of an affine connection if and only if the curvature operator associated to gis such that R(·,D)D= 0. Moreover, the scalar curvature of every deformed Riemannian extension vanishes and its Ricci operator is nilpotent, so a deformed Riemannian extension will be Einstein if and only if it is Ricci-flat. Modified Riemannian extensions Going further in the generalization of Riemannian extensions we find a new kind of metrics known as modified Riemannian extensions. These metrics involve two new elements Sand Twhich are (1,1)-tensor fields on the affine manifold M. Modified 18 1.4 Lie groups Riemannian extensions are split-signature metrics on the cotangent bundle of Mas well and are defined by gD,Φ,T,S =ιT ◦ιS +gD,Φ =ιT ◦ιS +gD+π∗Φ, where ◦denotes the symmetric product of one-forms. This kind of metrics can be expressed in terms of the local coordinates induced on the cotangent bundle as gD,Φ,T,S =1 2xr′xs′(TirSjs+TjrSis)−2xk′Γijk+ Φij Idn Idn0, where i, j = 1, . . . , n and i′=i+n. In the particular case where T=cId and S= Id, the metric is denoted by gD,Φ,c and it is a Walker metric for which the tensor Bij in (1.8) is a quadratic function of the fibre coordinates (xi′). The Walker distribution is given by D= ker π∗and its scalar curvature is a multiple (depending on the dimension of the manifold) of the parameter c. Modified Riemannian extensions can be characterized amongst the Walker metrics in terms of the covariant derivative of their curvature by ∇DR(·,D)D= 0 (see [1]). As a consequence, an even-dimensional Walker manifold admitting a parallel, degenerate distribution of maximal dimension is locally symmetric if and only if it is a suitable modified Riemannian extension. In addition to all this, modified Riemannian extensions turn out to be a source of examples of Einstein metrics. In fact, the modified Riemannian extension gD,Φ,c, with c= 0, is Einstein if and only if Φ = 4 c(n−1)Dρs. If (M, D)is flat, its cotangent bundle is not only Einstein but also para-Kähler with constant para-holomorphic sectional curvature, when equipped with a modified Riemannian extension [41]. We will see what this means in Part I. 1.4 Lie groups ALie group is a smooth manifold endowed with a group structure such that the operation in the group σ:G×G→Gand the inversion are differentiable. A group homomorphism that is differentiable as a map between two smooth manifolds is called aLie group homomorphism. A real Lie algebra is a vector space gendowed with a skew-symmetric bilinear operator [·,·]: g×g→gsatisfying the Jacobi identity [x, [y, z]] + [z, [x, y]] + [y, [z, x]] = 0 19 Preliminaries for all x, y, z ∈g. A Lie algebra homomorphism is a linear map φ:g→hthat preserves the Lie brackets, i.e., such that φ[x, y]g= [φ(x), φ(y)]hfor all x, y ∈g. Given a Lie algebra g, a vector subspace his a Lie subalgebra of gif his closed for the Lie brackets, i.e., if [x, y]∈hfor all x, y ∈h. An ideal aof gis a Lie subalgebra satisfying the stronger condition that [x, a]∈afor all x∈gand a∈a. A particular example of an ideal is the derived subalgebra of a given Lie algebra g, which is the subspace g′= span {[x, y]: x, y ∈g}and is also denoted by [g,g]. A Lie algebra such that the sequence of subalgebras given by g≥[g,g]≥[g,[g,g]] ≥[g,[g,[g,g]]] ≥. . . terminates in the zero subalgebra is said to be nilpotent. The Lie algebras whose derived subalgebras are nilpotent are called solvable, and those which have no nonzero solvable ideals are said to be semi-simple. Given a Lie group G, a vector field Xon Gis said to be left-invariant if for all g, h ∈G, (Lg)∗hXh=XLg(h), where Lg:h∈G→gh ∈Gdenotes the left translations on G. The Lie brackets induce a Lie algebra structure in the space of left-invariant vector fields on a Lie group G, which is called the Lie algebra of Gand is denoted by Lie(G) = g. The Lie algebra of a Lie group is isomorphic to the tangent space at the identity element of G. Besides, any Lie algebra of finite dimension gis isomorphic to the Lie algebra of a unique connected and simply connected Lie group – up to isomorphisms. This allows us to identify each simply connected Lie group with its Lie algebra. In the situation where the Lie group is also a pseudo-Riemannian manifold, the need to study metrics for which Lgis an isometry – known as left-invariant metrics – arises naturally. Giving a left-invariant metric on a Lie group Gis equivalent to giving an inner product on its Lie algebra g. The invariance of the metric allows us to obtain general expressions for the Levi-Civita connection, curvature, etc. of a pseudo-Riemannian Lie group just by knowing how the brackets and the metric behave. Semi-direct products Given two Lie algebras (h1,[·,·]1)and (h2,[·,·]2), if we consider the Lie algebra g=h1⊕h2given by the direct sum of h1and h2as vector spaces, we can take the Lie brackets [v, v′] = [v, v′]1,[w, w′] = [w, w′]2and [v, w] = 0 for all v, v′∈h1and w, w′∈h2. Then (g,[·,·]) is the direct sum Lie algebra. 20 1.4 Lie groups and so every semi-direct extension is of the form E(1,1) ⋊φRe0, where φbelongs to Der(e(1,1)). If we now rescale the vector e0and take ˆe0=e0−ae1+ce2−de3, and consider the basis of E(1,1) ⋊φRˆe0given by {e1, e2, e3,ˆe0}, then [ˆe0, e1] = 0,[ˆe0, e2] = 0,[ˆe0, e3] = (a+b)e3, so there are two different possibilities depending on whether or not a+b= 0. If a+b= 0, then E(1,1) ⋊φRˆe0is unimodular and its Lie brackets are [ˆe0, ei] = 0,[e1, e2] = e2,[e1, e3] = −e3. This corresponds to the product Lie group E(1,1) ×Rˆe0. If a+b= 0, then E(1,1)⋊φRe0is non-unimodular and its non-zero Lie brackets are [e1, e2] = e2,[e1, e3] = −e3,[e3,ˆe0] = λe3,(λ= 0). If we now consider the Lie group homomorphism Φgiven by Φ(e1) = −1 λˆe0,Φ(e2) = e3,Φ(e3) = e1−1 λˆe0,Φ(ˆe0) = λe2 and take the basis {v1,v2,v3,v4},vi= Φ(ei), of E(1,1) ⋊φRˆe0, then the Lie brackets are [v1,v2] = v2,[v3,v4] = v4 and this Lie algebra corresponds to aff(R)×aff(R), where aff(R)denotes the real affine Lie algebra. The discussion in the previous remark can be summarized as follows. Lemma 1.9. Let G=E(1,1) ⋊ R be a semi-direct extension of the Poincaré Lie group. Then (i) Gis unimodular if and only if it is isomorphic to the product E(1,1) ×R, and (ii) Gis non-unimodular if and only if it is isomorphic to aff(R)×aff(R). Remark 1.10 (The Euclidean Lie algebra e(2)).The Euclidean Lie group can be described as the semi-direct product R2⋊ψRgiven by ψ= ad(e1) = 0−1 1 0 , so its non-zero Lie brackets are given by [e1, e2] = e3,[e1, e3] = −e2. 27 Preliminaries The derivations of this Lie algebra are Der(e(2)) =   φ=  0 0 0 c a −b d b a  :a, b, c, d ∈R   and so every semi-direct extension is of the form e E(2)⋊φRe0, where φ∈Der(e(2)). We can rescale e0by ˆe0=e0−be1+de2−ce3and consider the basis of e E(2)×Rˆe0 given by {e1, e2, e3,ˆe0}so that [ˆe0, e1] = 0,[ˆe0, e2] = ae2,[ˆe0, e3] = ae3. This gives rise to two different possibilities depending on whether or not a= 0. If a= 0, then e E(2) ⋊φRˆe0is unimodular and its Lie brackets are [ˆe0, ei] = 0,[e1, e2] = e3,[e1, e3] = −e2. This corresponds to the product Lie group e E(2) ×Rˆe0. If a= 0, then the semi-direct product e E(2) ⋊φRˆe0is non-unimodular and its Lie algebra is isomorphic to [ˆe0, e2] = e2,[ˆe0, e3] = e3,[e1, e2] = e3,[e1, e3] = −e2, which corresponds to the Lie algebra aff(C)×aff(C), where aff(C)is the complex affine Lie algebra. The discussion above can be summarized as follows. Lemma 1.11. Let G=e E(2) ⋊ R be a semi-direct extension of the Euclidean Lie group. Then (i) Gis unimodular if and only if it is isomorphic to the product e E(2) ×R, and (ii) Gis non-unimodular if and only if it is isomorphic to aff(C)×aff(C). The left-invariant metrics on each four-dimensional solvable Riemannian Lie group can be described in terms of an orthonormal basis as follows. Left-invariant Riemannian metrics on H3⋊ R Let g=h3⋊ R be the semi-direct extension of the Heisenberg algebra h3. Let ⟨·,·⟩ be an inner product in gand let ⟨·,·⟩3be its restriction to h3. It follows from Milnor’s work [104] that there exists an orthonormal basis {v1,v2,v3}of h3such that [v1,v2] = γv3,[v1,v3] = 0,[v2,v3] = 0, γ = 0.(1.11) 28 1.4 Lie groups The algebra of derivations of h3with respect to a rotated basis that we will also denote by {v1,v2,v3}is given by Der(h3) =     ˜a˜c0 −˜c˜ d0 ˜ h˜ f˜a+˜ d  : ˜a, ˜c, ˜ d, ˜ h, ˜ f∈R  . Let {v1,v2,v3,v4}be a basis of gwith {v1,v2,v3}given by Equation (1.11), and g=h3⊕Rv4. Since Rv4need not be orthogonal to h3, we set ˜ ki=⟨vi,v4⟩, for i= 1,2,3, and consider ˆe4=v4−Pi˜ kivi. If we now normalize it we get an orthonormal basis {e1, e2, e3, e4}of gsuch that [e1, e2] = γe3,[e4, e1] = 1 R{˜ae1−˜ce2+ (˜ h+˜ k2γ)e3}, [e4, e3] = 1 R(˜a+˜ d)e3,[e4, e2] = 1 R{˜ce1+˜ de2+ ( ˜ f−˜ k1γ)e3}, (1.12) where R > 0. In order to simplify the expressions we define a=−˜a R, c =−˜c R, d =−˜ d R, h =−˜ h R, f =−˜ f R, k1=−˜ k1 R, k2=−˜ k2 R and use the notation F=f−k1γand H=h+k2γ. Now the Lie brackets in Equation (1.12) become [e1, e2] = γe3,[e1, e4] = ae1−ce2+He3, [e3, e4] = (a+d)e3,[e2, e4] = ce1+de2+Fe3.(1.13) Left-invariant Riemannian metrics on E(1,1) ⋊ R and e E(2) ⋊ R Let g=g3⋊ R be the semi-direct extension of the three-dimensional Lie algebra g3, being g3either e(1,1) or e(2). Let ⟨·,·⟩be an inner product in gand let ⟨·,·⟩3denote its restriction to g3. According to Milnor’s work [104], there exists an orthonormal basis {v1,v2,v3}of g3such that [v2,v3] = λ1v1,[v3,v1] = λ2v2,[v1,v2]=0,(1.14) for λ1, λ2∈Rand λ1λ2= 0. The associated Lie groups correspond to E(2) whenever λ1λ2>0and E(1,1) whenever λ1λ2<0. Moreover, the algebra of derivations of g3is given by Der(g3) =     ˜ b˜a˜c −λ2 λ1˜a˜ b˜ d 0 0 0  : ˜a,˜ b, ˜c, ˜ d∈R  . 29 Preliminaries Let {v1,v2,v3,v4}be a basis of gfor which ad(v4)is determined by a derivation as above. After a normalization, like in the previous section, we get an orthonormal basis {e1, e2, e3, e4}for which the non-zero Lie brackets are given by [e2, e3] = λ1e1,[e3, e1] = λ2e2, [e4, e1] = 1 R{˜ be1−λ2(˜a λ1+˜ k3)e2}, [e4, e2] = 1 R{(˜a+˜ k3λ1)e1+˜ be2}, [e4, e3] = 1 R{(˜c−˜ k2λ1)e1+ ( ˜ d+˜ k1λ2)e2} (1.15) where R > 0. In order to simplify the notation, we define a=−˜a R, b =−˜ b R, c =−˜c R, d =−˜ d R, k1=−˜ k1 R, k2=−˜ k2 R, k3=−˜ k3 R and set A=a λ1+k3,C=c−k2λ1and D=d+k1λ2. Now the Lie brackets given in Equation (1.15) become [e1, e3] = −λ2e2,[e2, e3] = λ1e1, [e1, e4] = be1−Aλ2e2,[e2, e4] = Aλ1e1+be2, [e3, e4] = Ce1+De2. (1.16) Left-invariant Riemannian metrics on R3⋊ R Let g=r3⋊R be a semi-direct extension of the Abelian Lie algebra r3. Let ⟨·,·⟩be a inner product on gand ⟨·,·⟩3be its restriction to r3. The algebra of all the derivations of r3is gl(3,R)and there exists a ⟨·,·⟩3-orthonormal basis {v1,v2,v3}of r3where a derivation decomposes as a sum of a diagonal matrix and a skew-symmetric matrix. Therefore, the algebra of derivations of r3is given by Der r3=    ˜a−˜ b−˜c ˜ b˜ f−˜ h ˜c˜ h˜p  : ˜a,˜ b, ˜c, ˜ f, ˜ h, ˜p∈R  . Now, the corresponding semi-direct product g=r3⋊ R is given by [v1,v2] = 0,[v4,v1] = ˜av1+˜ bv2+ ˜cv3, [v1,v3] = 0,[v4,v2] = −˜ bv1+˜ fv2+˜ hv3, [v2,v3] = 0,[v4,v3] = −˜cv1−˜ hv2+ ˜pv4, with respect to some basis {v1,v2,v3,v4}so that g= span{v1,v2,v3} ⊕ Rv4. Since Rv4need not be orthogonal to r3, we can consider ki=⟨vi,v4⟩, for all 30 1.4 Lie groups i= 1,2,3, and set ˆe4=v4−Pikivi. If we normalize it, we obtain an orthonormal basis {e1, e2, e3, e4}of gso that [e4, e1] = 1 R(˜ae1+˜ be2+ ˜ce3),[e4, e2] = 1 R(−˜ be1+˜ fe2+˜ he3), [e4, e3] = 1 R(−˜ce1−˜ he2+ ˜pe3), R > 0. (1.17) In order to simplify the notation, we define a=˜a R, b =˜ b R, c =˜c R, f =˜ f R, h =˜ h R, p =˜p R. Now the Lie brackets given in Equation (1.17) become [e1, e4] = ae1+be2+ce3,[e2, e4] = −be1+fe2+he3, [e3, e4] = −ce1−he2+pe3.(1.18) Non-solvable Lie groups Four-dimensional non-solvable Lie groups are isomorphic to either one of the direct products SU(2) ×Rand f SL(2,R)×Rand the left-invariant metrics on them can be described in terms of an orthonormal basis as follows. Left-invariant Riemannian metrics on SU(2) ×Rand f SL(2,R)×R Let g=g3×Rbe a direct extension of the unimodular Lie algebra g3=sl(2,R)or g3=su(2). Let ⟨·,·⟩ be an inner product on gand let ⟨·,·⟩3denote its restriction to g3. Following Milnor’s work [104], there exists an orthonormal basis {v1,v2,v3}of g3such that [v2,v3] = λ1v1,[v3,v1] = λ2v2,[v1,v1] = λ3v3,(1.19) where λ1,λ2,λ3∈Rand λ1λ2λ3= 0. Moreover, the associated Lie group corresponds to SU(2) if λ1,λ2and λ3have the same sign, and to SL(2,R)otherwise. Let {v1,v2,v3,v4}be a basis of gsuch that {v1,v2,v3}are given by Equation (1.19) and g=g3⊕Rv4. Since Rv4need not be orthogonal to g3, we consider ˜ ki=⟨vi,v4⟩, for i= 1,2,3. Let ˆe4=v4−Pi˜ kiviand normalize it to get an orthonormal basis {e1, e2, e3, e4}of g=g3⊕Rsuch that [e1, e2] = λ3e3,[e2, e3] = λ1e1, [e3, e1] = λ2e2,[e1, e4] = 1 R(˜ k3λ2e2−˜ k2λ3e3), [e2, e4] = 1 R(˜ k1λ3e3−˜ k3λ1e1),[e3, e4] = 1 R(˜ k2λ1e1−˜ k1λ2e2), (1.20) 31 Preliminaries where R > 0. In order to simplify the expressions, we define ki=˜ ki R, so the Lie brackets now take the form [e1, e2] = λ3e3,[e1, e3] = −λ2e2,[e2, e3] = λ1e1, [e1, e4] = k3λ2e2−k2λ3e3,[e2, e4] = k1λ3e3−k3λ1e1, [e3, e4] = k2λ1e1−k1λ2e2. (1.21) 1.5 Homogeneous spaces Roughly speaking, in pseudo-Riemannian geometry, homogeneity means that the geometry of a manifold is the same at each of its points. What this means is that for any two points in the manifold, there exists an isometry sending one to the other. At the same time, in affine geometry, homogeneity means that for any two points there exists an affine transformation sending one point to the other. It is important to be aware of the fact that a pseudo-Riemannian manifold may be affine homogeneous for its Levi-Civita connection but not necessarily homogeneous (see [92]). Riemannian homogeneous spaces A connected Riemannian manifold (M, g)is said to be homogeneous if its isometry group acts transitively on M, i.e, if for any two points p, q ∈Mthere exists an isometry φof (M, g)such that φ(p) = q. In this situation, the connected component of the identity of the isometry group also acts transitively on M. This definition of homogeneity is equivalent to the existence of a connected Lie group Gand a smooth map G×M−→ M (q, p)7−→ Lq(p) = q p such that (i) Lqis an isometry of (M, g). (ii) Lq1Lq2=Lq1q2. (iii) For any p1, p2∈M, there exists an element q1∈Gsuch that Lq1(p1) = p2. If Gacts effectively on M, i.e., if Lqis the identity transformation of Mif and only if qis the identity element e∈G, we can always replace Gby the quotient group G/K, where Kis the kernel of the map q∈G7→ Lq∈Isom(M). Therefore, if Gis a connected Lie group that acts on (M, g)as a transitive and effective group of 32 1.6 A note on Gröbner bases isometries, then Gcan be identified with a Lie subgroup of the isometry group of (M, g). Let p∈Mand H={q∈G:q p =p}be the isotropy group of p. Then Mis diffeomorphic to the quotient G/H and we have the canonical projection π:G−→ G/H. This gives a fibre bundle over Mwith structure group H, where the subgroup His closed, but not necessarily connected. A Riemannian metric ⟨·,·⟩ on G/H is said to be G-invariant if the action tq:sH ∈G/H 7−→ tq(sH) = q sH is an isometry for all q∈G. In this case (G/H, ⟨·,·⟩)is called a Riemannian homogeneous space.(M, g)is locally homogeneous if for each two points p, q ∈Mthere exist neighbourhoods Uof pand Vof qand a local isometry φ:U → V such that φ(p) = q. Simply connected homogeneous Riemannian manifolds of dimension two are symmetric. Three-dimensional simply connected homogeneous Riemannian manifolds are either symmetric spaces or Lie groups endowed with left-invariant Riemannian metrics (see [126], [103] for a modern presentation and [30] for an extension to the three-dimensional Lorentzian case). Bérard-Bergery showed in [15] that the same result holds true in the four-dimensional situation. Theorem 1.12. Let (M, g)be a four-dimensional simply connected Riemannian homogeneous manifold. Then, it is either symmetric or isometric to a Lie group endowed with a left-invariant metric. Let (M, g)be a connected n-dimensional Riemannian manifold. What is more, consider M=G/H, where Gis a subgroup of the group of isometries of Macting transitively and effectively on M, and His the isotropy group of a point p∈M. If we denote by gand hthe Lie algebras of Gand H, respectively, then M=G/H is said to be reductive if there exists a vector subspace mof gsuch that g=h⊕m, and mis the ad(H)-invariant subspace of g. 1.6 A note on Gröbner bases Gröbner bases were introduced by Bruno Buchberger around the year 1960 and they have proven themselves to be extraordinarily useful in many different mathematical 33 Preliminaries contexts. The aim of this section is to provide the reader with some basic knowledge on these algebraic objects, since we will be making use of them in different sections throughout this thesis as a tool to solve large system of polynomial equations. 1.6.1 Monomial order and ideals Given a monomial xα=xα1 1···xαn n, the exponents α= (α1, . . . , αn)are elements of Zn ≥0, which establishes a one-to-one correspondence between the monomials in the polynomial ring R[x1, . . . , xn]and Zn ≥0. A monomial order on R[x1, . . . , xn] is a relation >on Zn ≥0or, equivalently, on the set of monomials xα, satisfying the following properties. 1. >is a total order on Zn ≥0. 2. If α > β and γ∈Zn ≥0, then α+γ > β +γ. 3. >is a well-order on Zn ≥0. There are many different monomial orders, but we will be most interested in the following three: •Lexicographical order:α >lex βif the leftmost non-zero entry in the vector α−β∈Znis positive. •Graded lexicographical order:α >grlex βif |α|>|β|or |α|=|β|with α >lex β. •Graded reverse lexicographical order:α >grevlex βif |α|>|β|or |α|=|β| and the rightmost non-zero entry of α−β∈Znis negative. The lexicographical order corresponds to the alphabetical order and a variable dominates any monomial involving only smaller variables, regardless of its total degree. If we want to take into account the total degrees of the monomials so that the monomials of higher degree are the greatest, we can use the graded lexicographical order. Let P=Pαaαxαbe a non-zero polynomial in R[x1, . . . , xn]and let >be a monomial order. The multidegree of Pis the maximum (with respect to the monomial order >)α∈Zn ≥0so that aα= 0. The corresponding monomial is called the leading term LT(P) = aαxα. A monomial ideal in R[x1, . . . , xn]is a polynomial ideal that can be generated by monomials. A polynomial Pbelongs to a monomial ideal Iif and only if all of its terms are elements of I. We denote by LT (I)the set of leading terms of the non-zero elements of I, i.e., LT(I) = {cxα:∃P∈ I \{0}s.t. LT (P) = cxα}, 34 1.6 A note on Gröbner bases and ⟨LT(I)⟩denotes the ideal generated by the elements of LT(I). Notice that if Pi∈ I, for i= 1, . . . , k, then LT(Pi)∈LT(I)⊂ ⟨LT(I)⟩and so ⟨LT(P1), . . . , LT(Pk)⟩ ⊂ ⟨LT(I)⟩. However, if I=⟨P1, . . . , Pk⟩, then ⟨LT(I)⟩might be strictly larger than the ideal ⟨LT(P1), . . . , LT(Pk)⟩. Consider, for instance, the ideal I=⟨P1,P2⟩where P1=y3−2xy, P2=xy2−2x2+y, and fix the graded lexicographical order for monomials. The polynomial yP2−xP1=y2 belongs to Iand y2=LT(x2)∈ ⟨LT(I)⟩, but y2/∈ ⟨LT (P1), LT(P2)⟩=⟨y3, xy2⟩. Theorem 1.13 (Hilbert Basis Theorem).Every ideal I ⊂ R[x1, . . . , xn]has a finite generating set. The analogue result for monomial ideals is called Dickson’s Lemma. The Hilbert Basis Theorem guarantees that any non-zero ideal I ⊂ R[x1, . . . , xn]admits a Gröbner basis. Definition 1.14. A finite subset G={g1, . . . gν}of an ideal Iwith a fixed monomial order such that ⟨LT(g1), . . . , LT(gν)⟩=⟨LT(I)⟩ is said to be a Gröbner basis (or a Gröbner-Shirshov basis) with respect to the given monomial order. Gröbner bases provide us with a powerful tool to find quite simple algorithmic solutions to various algebraic problems. For example: The Ideal Membership Problem The remainder of the division algorithm applied to a polynomial Pdivided by the elements of a Gröbner basis Gof an ideal Iis zero if and only if Pbelongs to I, and this property does not necessarily hold if Gis not a Gröbner basis. Solving large systems of polynomial equations Let {Pi}be a set of polynomials in R[x1, . . . , xn]and consider the system of polynomial equations given by {Pi= 0}. If the system in consideration is simple, it is an elementary problem to find all the common roots, but if the number of 35 Preliminaries unknowns, equations and their degree increase, then finding all the solutions might become quite an unmanageable problem. What can one do to make the task easier? If two sets of polynomials generate the same ideal, the corresponding zero sets must be identical. The theory of Gröbner bases provides a well-known strategy to solve rather large polynomial systems obtaining “better” polynomials that belong to the ideal generated by the initial polynomial system. We refer to [53] for more information regarding the theory of Gröbner bases. 1.6.2 Buchberger’s algorithm Buchberger’s algorithm is the oldest algorithm ever introduced for the computation of Gröbner bases. It was devised by Bruno Buchberger at the same time that he introduced Gröbner bases. A crude version of this algorithm to find a Gröbner basis of an ideal of polynomials Iproceeds as follows: Input: A set of polynomials Pthat generate I. Output: A Gröbner basis Gfor I. (1) Set G=P. (2) If pi, pj∈Pand we denote by fi, fjthe coefficients of their leading terms with respect to a given monomial ordering, respectively, set aij = lcm{fi, fj}. (3) Define gij =aij fipi−aij fjpj. Note that the leading terms here will cancel. The polynomials piand pjare called a critical pair. (4) Reduce gij as much as possible using the multivariate division algorithm with respect to G. If the result is non-zero, then add gij to G. (5) Repeat (2)–(4) until all possible pairs have been considered, including those involving the new polynomials added to Gin step (4). There are many ways in which the algorithm above can be improved and, in fact, has been improved (see, for instance, [64,65]). 36 Null-Kähler structures A(1,1)-tensor field Jon a 4n-dimensional manifold Mis said to be an almost tangent structure if J2= 0. If, in addition, rank(J) = 2n,Jis said to be a null structure on M. A metric tensor gon Mis said to be null-Hermitian if g(JX, Y ) = g(X, JY ) for all vector fields X,Yon M, where Jis a null structure on M. This implies that g(X, JX) = 0. The signature of a null-Hermitian metric on Mis neutral (2n, 2n)and to each null-Hermitian metric there is an associated two-form defined by Ω(X, Y ) = g(JX, Y ). The kernel of a null structure is an integrable distribution of the tangent bundle and this integrability condition is equivalent to the vanishing of the associated Nijenhuis tensor given by (1.22). A null-Hermitian manifold (M, g, J)is said to be null-Kähler if ∇J= 0, where ∇denotes de Levi-Civita connection of g. The fundamental two-form given by Ω(X, Y ) = g(JX, Y )is covariantly-constant and so it is closed. Besides, it satisfies Ω∧n= Ω ∧ ··· ∧ Ω= 0 and Ω∧(n+1) = 0, in contrast with the Kähler and para-Kähler conditions, where Ω∧2n= 0. We refer to Dunajski’s works [60, 61] for more information about null-Kähler structures. Notation In what follows, (M, g, Jε)with J2 ε=−εId, g(JεX, Y ) + g(X, JεY)=0,∇Jε= 0, will denote a Kähler, a para-Kähler or a null-Kähler manifold for ε= 1,ε=−1, or ε= 0, respectively. 43 Chapter 2 Locally conformally flat four-dimensional structures The contents of this chapter regarding locally conformally flat Kähler and paraKähler surfaces are contained in the work [68]. 2.1 Locally conformally flat four-dimensional Kähler and para-Kähler structures Let (M, g, Jε)be a (para-)Kähler manifold, where J2 ε=−εId is the (para-)complex structure satisfying g(JεX, JεY) = εg(X, Y )for ε=±1. Since the (para-)complex structure is parallel, the curvature identity R(X, Y )·Jε=Jε·R(X, Y ), which strictly restricts the curvature tensor, holds. A consequence of this is that the Ricci tensor of a2n-dimensional locally conformally flat (para-)Kähler manifold satisfies (2n−4) ρ(X, Y ) = −τ 2n−1g(X, Y ) and the manifold is flat if nis greater than or equal to three. In addition, the Ricci tensor of a locally conformally flat (para-)Kähler manifold is parallel an so (M, g)is locally symmetric in dimension four [111, 129]. Tanno proved in [129] that locally conformally flat positive definite four-dimensional Kähler manifolds are either flat or a product of two surfaces of constant opposite curvature. This result does not cover all the possibilities in the pseudo-Riemannian case of split signature, as Patterson pointed out in [119]. Motivated by this and some recent interest in locally conformally flat (para-)Kähler surfaces [2,4,82], in this chapter we will give the complete classification of locally conformally flat (para-)Kähler surfaces showing the existence of two additional possibilities. The result that completes the classification in the paraKähler setting is given by Theorem 2.1, which is the main result in this chapter. Theorem 2.1. Any indecomposable locally conformally flat para-Kähler surface (M, g, J−)is locally isometric to the cotangent bundle T∗Σof a flat affine surface (Σ, D)with a para-complex structure determined by J|ker π∗= Id, where πdenotes the canonical projection from the cotangent bundle, and the metric gis given by g=ιT ◦ιId +gDwhere 45 Locally conformally flat four-dimensional structures (i) Tis a parallel nilpotent (1,1)-tensor field on (Σ, D), or (ii) Tis a parallel (1,1)-tensor field on (Σ, D)satisfying T2=−κ2Id. Moreover, in both cases the para-Kähler two-form Ω−(X, Y ) = g(J−X, Y )is the canonical symplectic form of the cotangent bundle. Recall that a pseudo-Riemannian manifold is indecomposable if it does not admit a non-degenerate subspace that is invariant under the action of its holonomy group. Besides, the holonomy group may act indecomposably without acting irreducibly. Para-Kähler surfaces in Case (i) in Theorem 2.1 had already been reported by Patterson in [119], while Case (ii) seems to be missing in previous works. Considering the Ricci operator in Case (ii),1 κRic defines a self-adjoint complex structure that is parallel, and so it is a Riemannian complex structure. Since the Ricci operator and the para-complex structure J−commute with each other, one has that J+=1 κRic J−is a complex structure so that (g, J+)is a locally conformally flat Kähler structure. Therefore, the result that completes the classification in the Kähler setting is given by Theorem 2.2. Theorem 2.2. Any indecomposable locally conformally flat Kähler surface (M, g, J+)is locally isometric to the cotangent bundle (T∗Σ, g)of a Riemannian surface (Σ, gΣ)of constant curvature with a metric given by (i) g=g∇Σif the Gaussian curvature is non-zero, or (ii) g=ιJΣ◦ιId +g∇Σif the Gaussian curvature vanishes, where ∇Σis the Levi-Civita connection of (Σ, gΣ)and JΣis the Kähler structure on Σassociated to the Riemannian volume form. Furthermore, the complex structure J+on T∗Σis determined by the symplectic form Ω+=−dιJΣ. Remark 2.3.In Chapter 3 we will see that it is possible to give a description of the structures in Theorems 2.1 and 2.2 in terms of left-invariant metrics on Lie groups. The metric tensors in (ii) in both Theorem 2.1 and Theorem 2.2 are the same, but their associated symplectic structures and underlying geometries are different. The metrics in Case (i) in both theorems above correspond to different curvature models that will be described in Section 2.1.1. The Kähler metrics in Theorem 2.2-(i) are locally isometric (up to reversing the metric) to those studied by Guilfoyle and Klingenberg on the space of oriented affine lines in R3by means of the minitwistor correspondence in [82]. Analogously, the metrics in Theorem 2.1-(i) are locally isometric (up to reversing the metric) to the space of oriented spacelike or timelike lines in R3 1(see [2, 4]). The metrics 46 2.1 Locally conformally flat four-dimensional Kähler and para-Kähler structures corresponding to Assertion (ii) in Theorem 2.1 and Theorem 2.2 are locally isometric (up to reversing the metric) to the non-Einstein para-Kähler and Kähler metrics in the space of spacelike and timelike oriented geodesics of the de Sitter space constructed by Anciaux in [4]. Besides, the non-locally conformally flat Kähler-Einstein metrics on the de Sitter space constructed in [4] correspond to those in Remark 2.13. 2.1.1 Curvature models In this section we will work at a purely algebraic level to describe the curvature models found in our description of locally conformally flat Kähler and para-Kähler surfaces. Bearing this in mind, let (V,⟨·,·⟩, J±)be a (para-)Hermitian inner product space. Recall that an algebraic curvature tensor is a multilinear map A:V ×V ×V ×V → R satisfying the identities (1.1). If the corresponding Weyl curvature tensor vanishes, then the algebraic curvature tensor Ais determined by the associated Ricci tensor and a straightforward calculation shows that the Ricci operator of a locally conformally flat (para-)Kähler surface is either diagonalizable or (following the discussion in [112]) its Jordan normal form corresponds to one of the following. 1. The Ricci operator has two 2×2Jordan blocks. At each point there is a basis {u1, v1, u2, v2}of the tangent space so that the Ricci operator and the non-zero inner products are given by Ric =     0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0    ,⟨ui, vi⟩=εi, ε2 i= 1 (i= 1,2). Besides, if the associated Weyl curvature vanishes, there are two different possibilities as ε1ε2=±1(up to reversing the metric). 1.a. If ε1ε2= 1, there is a unique (up to sign) Ricci-commuting Hermitian structure (⟨·,·⟩, J+)given by J+u1=−u2, J+v1=−v2 and there are no Ricci-commuting para-Hermitian structures. 1.b. If ε1ε2=−1, there is a unique (up to sign) Ricci-commuting paraHermitian structure (⟨·,·⟩, J−)given by J−u1=v2, J−v1=u2 and there are no Ricci-commuting Hermitian structures. 47 Locally conformally flat four-dimensional structures 2. The Ricci operator is complex diagonalizable with imaginary eigenvalues ±iκ. At each point there is a basis {u1, v1, u2, v2}of the tangent space so that the Ricci operator and the non-zero inner products are given by Ric =     0κ0 0 −κ0 0 0 0 0 0 κ 0 0 −κ0    ,⟨ui, ui⟩= 1 = −⟨vi, vi⟩,(i= 1,2). One may assume that κ > 0so that there are a unique (up to reversing the metric) Ricci-commuting para-Hermitian structure (⟨·,·⟩, J−)given by J−u1=v2, J−v1=−u2 and a unique (up to reversing the metric) Ricci-commuting Hermitian structure (⟨·,·⟩, J+)such that J+u1=u2, J+v1=−v2. Regarding the discussion above, we introduce the following locally conformally flat algebraic curvature models (V,⟨·,·⟩, A)given by A=1 2⟨·,·⟩⊙ρA, where ρAdenotes the Ricci tensors corresponding to the Ricci operators above and ⊙is the Kulkarni– Nomizu’s product. (M+) : (V,⟨·,·⟩, A)given by A1413 =A3231 =1 2 with respect to a basis {u1, u2, u3, u4}for which the non-zero inner products are ⟨u1, u2⟩= 1 = ⟨u3, u4⟩. (M−) : (V,⟨·,·⟩, A)given by A1413 =A3231 =−1 2 with respect to a basis {u1, u2, u3, u4}for which the non-zero inner products are ⟨u1, u2⟩= 1 = −⟨u3, u4⟩. (Nκ) : (V,⟨·,·⟩, A)given by A1413 =A1442 =A3224 =A3231 =κ 2 48 2.1 Locally conformally flat four-dimensional Kähler and para-Kähler structures with respect to a basis {u1, u2, u3, u4}where u1and u3are spacelike vectors and u2 and u4are timelike vectors. Notice that, even though the curvature models (Nκ)are not isometric, they are all homothetic to the curvature model (N1). We will see that the curvature tensors of the locally conformally flat para-Kähler manifolds in Assertions (i) and (ii) in Theorem 2.1 are modelled on (M−)and (Nk), respectively. The curvature tensors of the locally conformally flat Kähler manifolds in Assertions (i) and (ii) in Theorem 2.2 are modelled on (M+)and (Nk), respectively. 2.1.2 Self-dual Walker manifolds Let (M, g, D)be a four-dimensional Walker manifold, i.e., a pseudo-Riemannian manifold of split signature (M, g)admitting a parallel, degenerate plane field Dof maximal dimension. As we have already mentioned in Section 1.3.1, there exist local coordinates x1, x2, x1′, x2′so that the Walker distribution is D= span ∂x1′, ∂x2′ and the metric takes the form g=dxi⊗dxi′+dxi′⊗dxi+gij x1, x2, x1′, x2′dxi⊗dxj.(2.1) The existence of a two-dimensional degenerate distribution Don a split-signature four-dimensional manifold (M, g)naturally induces an orientation. Let {u, v}be a basis of Dpfor p∈M, and denote by u∗and v∗their corresponding dual forms. The Hodge-star operator satisfies ⋆(u∗∧v∗) = ±(u∗∧v∗)and so, any four-dimensional Walker manifold is naturally oriented by the self-duality of u∗∧v∗. Considering local coordinates as in (2.1), the Walker orientation determined by ⋆(dx1′∧dx2′) = dx1′∧dx2′ corresponds to the volume element volg=dx1∧dx2∧dx1′∧dx2′. Self-dual Walker manifolds have been described by Calviño-Louzao, García-Río, Gilkey and Vázquez-Lorenzo as follows. Theorem 2.4. ( [41], Theorem 7.1) A four-dimensional Walker manifold is self-dual if and only if it is locally isometric to the cotangent bundle T∗Σof an affine surface (Σ, D)with metric g=ιX (ιId ◦ιId) + ιT ◦ιId +gD+π∗Φ(2.2) where Xis a vector field on Σand Tand Φare a (1,1)-tensor field and a symmetric (0,2)-tensor field on Σ, respectively. A special case of this result describes the local structure of para-complex space forms as follows. 49 Locally conformally flat four-dimensional structures Theorem 2.5. ( [41], Theorem 2.2) A para-Kähler surface of non-zero constant paraholomorphic sectional curvature cis locally isometric to the cotangent bundle of a flat affine surface equipped with the modified Riemannian extension g=c ι Id ◦ιId +gD. Consider θ(p,ω)=π∗ωp=xℓ′dxℓthe tautological one-form of T∗Σand let Ω = dθ =dxℓ′∧dxℓ be the canonical symplectic form of T∗Σ. Given the modified Riemannian extension g=c ι Id ◦ιId +gD, one naturally has a para-complex structure J−determined by Ω (X, Y ) = g(J−X, Y ), whose components are J−∂xi′=∂xi′, J−∂xi=−∂xi+cxi′xj′∂xj′, and the Walker distribution D= ker π∗corresponds to the eigenspace D+= ker (J−−Id) . 2.1.3 Locally symmetric self-dual Walker surfaces Given that locally conformally flat Kähler surfaces with non-diagonalizable Ricci operator and locally conformally flat para-Kähler surfaces are locally symmetric, we study first which self-dual Walker surfaces in Theorem 2.4 are locally symmetric. Let Dρdenote the Ricci tensor of (Σ, D)and decompose it as Dρ=Dρs+Dρsk, where Dρsand Dρsk denote the symmetric and the skew-symmetric parts of Dρ, respectively. Lemma 2.6. Let (M, g)be a locally symmetric self-dual Walker manifold. Then the Riemannian extension gin Theorem 2.4 satisfies g=ιT ◦ιId +gD+π∗Φfor some parallel (1,1)-tensor field Ton (Σ, D). Furthermore, if the scalar curvature is not zero, then (M, g)is locally isometric to a para-complex space form as in Theorem 2.5. Proof. The scalar curvature of any Riemannian extension in Theorem 2.4 is given by τ= 3 tr T+ 12ιX. Since the scalar curvature is constant, then X= 0 and tr T=κfor some κ∈R. Given the fact that any self-dual Walker manifold is locally isometric to a Riemannian extension given by Theorem 2.4, the covariant derivatives of the curvature operator are polynomials on the fibre coordinates (x1′, x2′). (∇∂x1R)(∂x2, ∂x1, ∂x1, ∂x1) = −1 8κT122x3 2′+other terms, (∇∂x2R)(∂x2, ∂x1, ∂x1, ∂x1) = 1 4κT212x3 1′+other terms, (∇∂x2R)(∂x2′, ∂x1, ∂x1, ∂x1) = −κ 8κ−2T22x1′+other terms. 50 2.1 Locally conformally flat four-dimensional Kähler and para-Kähler structures Assume in the first place that the scalar curvature τ= 3κ= 0. If ∇R= 0 then the previous expressions show that the tensor field Tmust be a scalar multiple of the identity, T=cId. Further calculations now show that (∇∂x1R)(∂x2, ∂x1, ∂x1, ∂x1) = c{2Dρs(∂x1, ∂x1)−1 2cΦ11}x2′+other terms, (∇∂x2R)(∂x2, ∂x1, ∂x1, ∂x1) = 3 2c{2Dρs(∂x1, ∂x2)−1 2cΦ12}x2′+other terms, (∇∂x2R)(∂x2, ∂x1, ∂x1, ∂x1′) = 1 2c2{2Dρs(∂x2, ∂x2)−1 2cΦ22}x3 1′+other terms, from where it follows that the symmetric (0,2)-tensor field Φ = 4 cDρs. In this situation, the Ricci operator Ric = 3c 2Id and, according to [41, Theorem 2.1], the corresponding metric is Einstein. Furthermore, one has that for any unit vector field X, the Jacobi operators RX=R(·, X)Xhave eigenvalues {0, c, 1 4c, 1 4c}and the eigenspace associated to cis timelike. Consequently, (M, g)is locally a paracomplex space form (see [77]), thus being locally isometric to a modified Riemannian extension given by Theorem 2.5. Now assume that the scalar curvature τ= 0. Let DT denote the covariant derivative of the (1,1)-tensor field Twith respect to the affine connection D. We set DT =DTj;ikdxi⊗dxj⊗∂xk, where DTj;ik=∂xiTjk+TjℓDΓiℓk. Since the scalar curvature vanishes, tr T= 0. Hence T11=−T22and thus DT1;11=−DT2;12and DT1;21=−DT2;22. A straightforward calculation now shows that (∇∂x1R)(∂x2′, ∂x1, ∂x1, ∂x1) = 1 2DT1;12, (∇∂x1R)(∂x2′, ∂x2, ∂x2, ∂x1) = 1 2DT2;11, (∇∂x2R)(∂x2′, ∂x2, ∂x1, ∂x2) = 1 2DT1;22, (∇∂x1′R)(∂x1, ∂x2, ∂x2, ∂x1) = −1 2DT2;21, (∇∂x1′R)(∂x1, ∂x2, ∂x1, ∂x1) = DT2;22+1 2DT2;11, (∇∂x2′R)(∂x1, ∂x2, ∂x2, ∂x2) = DT2;12−1 2DT1;22, from where it follows that the trace-free (1,1)-tensor field Tis D-parallel. The existence of a parallel (1,1)-tensor field on an affine surface (Σ, D)was considered in [40] showing that (besides the trivial case where T= 0) a parallel trace-free (1,1)-tensor field corresponds to one of the following: (a) An affine para-Kähler structure (det T=−k2<0), which in suitable adapted coordinates becomes T=k(∂x1⊗dx1−∂x2⊗dx2). 51 Locally conformally flat four-dimensional structures (b) An affine nilpotent Kähler structure (T2= 0), which in suitable adapted coordinates becomes T=k∂x1⊗dx2. (c) An affine Kähler structure (det T=k2>0), which in suitable adapted coordinates becomes T=k(∂x2⊗dx1−∂x1⊗dx2). Each of the three possibilities above gives rise to different geometric structures which are locally conformally flat. We study each case separately in what follows. Locally symmetric self-dual Walker surfaces given by an affine para-Kähler structure We will see that this case leads to product manifolds. Lemma 2.7. Let (T∗Σ, g)be a locally symmetric self-dual Walker manifold determined by an affine para-Kähler structure Ton (Σ, D). Then (T∗Σ, g)is locally conformally flat and locally isometric to a product of two Lorentzian surfaces of constant opposite Gaussian curvature. Proof. Let (Σ, D)be an affine surface and choose local coordinates (x1, x2)so that the parallel tensor field Tis locally given by T=k(∂x1⊗dx1−∂x2⊗dx2). Then Tis parallel if and only if the Christoffel symbols are such that DΓ112=DΓ121=DΓ122=DΓ221= 0. We refer to [40] for more information on the matter. Let (x1, x2, x1′, x2′)be the induced coordinates on T∗Σ. Then the symmetric and skew-symmetric Ricci tensors of (Σ, D)are given by Dρs=−(∂x1DΓ222+∂x2DΓ111)dx1◦dx2, Dρsk =1 2(∂x1DΓ222−∂x2DΓ111)dx2∧dx1. A straightforward calculation now shows that the Ricci operator of (T∗Σ, g), when expressed on the coordinate basis, satisfies Ric =     k0 0 0 0−k0 0 0kΦ12 + 2Dρs(∂x1, ∂x2)k0 −kΦ12 + 2Dρs(∂x1, ∂x2) 0 0 −k    , from where it follows that the Ricci curvatures are ±k. Moreover, a straightforward calculation shows that (Ric −kId)(Ric +kId) = 0, so the Ricci operator is diagonalizable with respect to an orthonormal basis. If it is parallel, then the manifold is locally isometric to a product of two Lorentzian surfaces of constant opposite Gaussian curvature, thus being locally conformally flat. 52 2.1 Locally conformally flat four-dimensional Kähler and para-Kähler structures for some function h(x1, x2)and the almost para-complex structure becomes Jh −∂x1=−∂x1+g11∂x1′+ (g12 +h)∂x2′, Jh −∂x1′=∂x1′, Jh −∂x2=−∂x2+ (g12 −h)∂x1′+g22∂x2′, Jh −∂x2′=∂x2′.(2.5) Let (g, Jh −)be an almost para-Hermitian structure determined by (2.3) and (2.5). The para-Kähler two-form is given by Ωh=h dx1∧dx2+dx1′∧dx1+dx2′∧dx2. Notice that the para-Kähler orientation and the Walker orientation are opposite. Indeed, the para-Kähler two-form Ωhis anti-self-dual for the para-Kähler orientation determined by the para-complex structure Jh −, but it is self-dual for the Walker orientation. With the intention of describing all the anti-self-dual para-Kähler surfaces of constant scalar curvature we consider the cotangent bundle T∗Σof an affine surface (Σ, D)with the metric g=ιT ◦ιId +gD+π∗Φas discussed in Section 2.1.3 and set the para-complex structure satisfying the condition J−|ker π∗= Id. The almost para-Hermitian structures (g, Jh −)defined by (2.3) and (2.5) are not para-Kähler in general. In order to express the components of ∇Jh −on T∗Σwe use the notation (∇∂xαJh −)∂xβ= (∇Jh −)β;αγ∂xγand (D∂xiΦ)(∂xj, ∂xk) = DΦjk;ito represent the covariant derivative of the symmetric (0,2)-tensor field Φon Σ. In this notation, the components of the covariant derivative of the para-complex structures Jh −are given by the following result. Lemma 2.16. Let (T∗Σ, g)be a locally symmetric self-dual Walker structure on the cotangent bundle of an affine surface (Σ, D). Let Jh −be a (locally defined) almost para-complex structure determined by Jh −|ker π∗= Id so that (g, Jh −)is an almost para-Hermitian structure locally given by (2.5). Then the non-zero components of ∇Jh −are determined by 2∇Jh −1;1 2′=1 4x2 1′x2′(T22)2−(T11)2−1 2x2′8Dρ(∂x1, ∂x1)−4hT12 +1 2x1′8Dρ(∂x2, ∂x1)+5hT11+hT22 +2(T21Φ11 −T12Φ22 + (T22−T11)Φ12) + 2 ∂1h−h(DΓ111+DΓ122) + DΦ11;2 −DΦ12;1, 2(∇Jh −)1;22′=1 4x1′x2 2′(T22)2−(T11)2+1 2x1′8Dρ(∂x2, ∂x2)+4hT21 −1 2x2′8Dρ(∂x1, ∂x2)−hT11−5hT22 −2(T21Φ11 −T12Φ22 + (T22−T11)Φ12) + 2 ∂2h−h(DΓ222+DΓ121) + DΦ12;2 −DΦ22;1, 59 Locally conformally flat four-dimensional structures where Tis a trace-free parallel (1,1)-tensor field on (Σ, D)and Φis a symmetric (0,2)-tensor field on Σ. Theorem 2.1 follows immediately from the following result describing the local structure of anti-self-dual para-Kähler surfaces with constant scalar curvature. Theorem 2.17. Let (M, g, J−)be an anti-self-dual para-Kähler surface with constant scalar curvature. Then it is locally isometric to a Riemannian extension of the form (T∗Σ,˜g=ιT ◦ιId +gD)with para-complex structure determined by J−|ker π∗= Id, where Tis a parallel (1,1)-tensor field on a flat affine surface (Σ, D) satisfying one of the following conditions. (i) T=cId and (M, g, J−)has constant para-holomorphic sectional curvature c. (ii) T= 0 and (M, g, J−)is flat. (iii) T2=k2Id and (M, g, J−)is locally isometric to a product of two Lorentzian surfaces of constant opposite curvature. (iv) T2= 0 and (M, g, J−)is modelled on (M−). (v) T2=−k2Id and (M, g, J−)is modelled on (Nk). In all the cases above the para-Kähler two-form is the canonical symplectic two-form of T∗Σ. Proof. Let (M, g, J−)be an anti-self-dual para-Kähler surface. Then there exists a Walker structure (M, g, D)so that (M, g)is self-dual with respect to the Walker orientation and (M, g, J−)is locally isometric to the cotangent bundle of an affine surface (Σ, D)with para-complex structure determined by J−|ker π∗= Id and metric tensor g=ιX(ιId ◦ιId) + ιT ◦ιId +gD+π∗Φ. A para-Kähler surface is anti-self-dual if and only if its Bochner tensor vanishes (see [27]), and it is locally symmetric if and only if the scalar curvature is constant. Assertion (i) corresponds to the case when its scalar curvature is non-zero and Lemma 2.6 shows that the (1,1)-tensor field Tis parallel in this case. Anti-self-dual para-Kähler surfaces whose scalar curvature is zero are locally conformally flat and locally symmetric. Therefore, the underlying structure is induced by an affine para-Kähler structure, an affine nilpotent Kähler structure, or an affine Kähler structure as discussed in Lemma 2.7, Lemma 2.9, and Lemma 2.11, respectively. Let (Σ, D, T)be an affine surface equipped with a parallel trace-free (1,1)-tensor field T. It follows from Lemma 2.16 that, if the almost para-complex structure Jh −determined by Jh −|ker π∗= Id is parallel, then it is uniquely determined. If 60 2.1 Locally conformally flat four-dimensional Kähler and para-Kähler structures (Σ, D, T)is an affine para-Kähler surface, then the coefficients of x1′and x2′in Lemma 2.16 show that h=−2 kDρs(∂x1, ∂x2)for a deformation tensor field Φgiven as in Remark 2.8. If (Σ, D, T )is an affine nilpotent Kähler surface, then Lemma 2.16 shows that h=−2 kDρ(∂x2, ∂x2)and if (Σ, D, T)is an affine Kähler surface, then h=2 kDρ(∂x2, ∂x2). Moreover, a straightforward calculation shows that for any (Σ, D, T)there is an appropriate deformation tensor field Φso that (T∗Σ, ιT ◦ιId +gD+π∗Φ, Jh −)is para-Kähler, where Φis given as in Remark 2.8, Remark 2.10 and Remark 2.12. In all these cases, (T∗Σ, ιT ◦ιId +gD+π∗Φ, Jh −)is locally isometric to the Riemannian extension ιT ◦ιId +gDof a flat affine surface (Σ, D, T)that is affine para-Kähler, affine nilpotent Kähler or affine Kähler, and the two-form of the corresponding locally conformally flat para-Kähler manifold is the canonical symplectic form of T∗Σ, from where Assertions (iii),(iv) and (v) follow. Finally, we consider the case T= 0 corresponding to Assertion (ii). Setting T= 0 in Lemma 2.16, one has that the non-zero components of ∇Jh −are given by ∇Jh −1;1 2′= 2x1′ρD 21 −2x2′ρD 11 +∂1h−h(DΓ111+DΓ122) +DΦ11;2 −DΦ12;1}, (∇Jh −)1;22′= 2x1′ρD 22 −2x2′ρD 12 +∂2h−h(DΓ222+DΓ121) +DΦ12;2 −DΦ22;1}. It now follows from the coefficients of the terms of degree one above that if ∇Jh −= 0, the Ricci tensor Dρvanishes and (Σ, D)is flat. Since the Ricci tensor of gD+π∗Φis determined by the symmetric part of Dρone has that (T∗Σ,˜g=gD+π∗Φ) is Ricciflat. Therefore, if (T∗Σ,˜g=gD+π∗Φ) is para-Kähler, then it must be Ricci-flat and thus flat – since it is locally conformally flat. Observe that in all the cases above the para-complex structure Jh −is uniquely determined because h= 0 if the base surface is flat and T= 0 (which follows in all cases from the expressions in Lemma 2.16). Moreover, the corresponding paraKähler form is again the canonical symplectic two-form of the cotangent bundle. Remark 2.18.The Ricci operator of any metric in Assertion (v) of Theorem 2.17 satisfies Ric2=−k2Id and, since the para-complex structure J−commute with the Ricci operator, defining J+=1 kRic ·J−one has that (g, J+)is a locally conformally flat indefinite Kähler structure. 61 Locally conformally flat four-dimensional structures 2.2 Locally symmetric Kähler surfaces Let (M, g, J+)be a locally symmetric four-dimensional Kähler manifold. Then its Ricci operator is parallel. In the diagonalizable case the metric is Einstein or locally isometric to a product of two surfaces of constant curvature. The non-diagonalizability of the Ricci operator leads to a Walker structure and therefore to the situation in Section 2.1.3. Lemma 2.19. A Kähler surface (M, g, J+)with parallel and non-diagonalizable Ricci operator is a Walker manifold. Proof. Since the Ricci operator Ric commutes with the complex structure J+, then the trace-less Ricci operator Ric0= Ric −τ 4Id is either two-step nilpotent or complex diagonalizable. If Ric0is two-step nilpotent, since ker Ric0is J+-invariant, then it must be twodimensional and parallel thus determining a totally degenerate parallel distribution, which shows that (M, g)is a Walker manifold. If Ric0is complex diagonalizable with eigenvalues ±ik, then 1 kRic0is a selfadjoint complex structure so that (M, g, 1 kRic0)is complex Riemannian and the (1,1)-tensor field J−=1 kRic0·J+determines a para-Kähler structure (g, J−)so that (M, g, J−)is a Walker manifold whose parallel distribution is J±-invariant. The proof of Theorem 2.2. Assertion (ii) in Theorem 2.2 follows immediately from Remark 2.18, which shows that any affine Kähler structure Tsatisfying T2=−k2Id on a flat affine surface (Σ, D)induces a Kähler structure (g, J+)on T∗Σwith g=ιT ◦ιId +gD and J+=−1 kRic ·J−, where J−is the para-Kähler structure determined by J−|ker π∗= Id . Moreover, let gΣbe a flat Riemannian metric on Σwith Levi-Civita connection D. A straightforward calculation shows that the corresponding Kähler two-form is given by Ω+=−dιJΣ, where JΣis the complex structure on (Σ, gΣ)induced by the volume element of the flat metric gΣ. Choosing local adapted coordinates (x1, x2)on Σso that the metric tensor gΣ=dx1⊗dx1+dx2⊗dx2, one has that the complex structure J+on T∗Σis characterized by J+∂x1′=∂x2′, J+∂x2′=−∂x1′, thus being a proper complex structure in the sense of [102] whose corresponding Kähler form is Ω+=dx1∧dx2′−dx2∧dx1′. 62 2.2 Locally symmetric Kähler surfaces We will construct a locally conformally flat Kähler surface showing the geometric realizability of the model (M+)and thus proving Assertion (i) in Theorem 2.2. Let (Σ, gΣ)be a Riemannian surface of non-zero constant Gaussian curvature and let Dbe its Levi-Civita connection. The Riemannian extension (T∗Σ, gD) is a locally symmetric four-dimensional manifold with curvature tensor modelled on (M+). Let ωΣbe the Riemannian volume form of (Σ, gΣ)and let JΣbe the complex structure gΣ(JΣX, Y ) = ωΣ(X, Y ). Then Ω+=−dιJΣis a symplectic structure on (T∗Σ, g =gD)which induces a Kähler structure with complex structure gD(J+ξ, η)=Ω+(ξ, η)for all vector fields ξand ηon T∗Σ. Let (x1, x2)be a system of local coordinates on Σso that the metric takes the form gΣ= Ψ(x1, x2)(dx1⊗dx1+dx2⊗dx2)and JΣ∂x1=∂x2, JΣ∂x2=−∂x1. Then the complex structure J+is determined, with respect to the induced system of local coordinates (x1, x2, x1′, x2′), by J+∂x1′=∂x2′, J+∂x2′=−∂x1′. It corresponds to a proper Kähler structure whose corresponding Kähler form is given by Ω+=dx1∧dx2′−dx2∧dx1′. Remark 2.20.Let Σ = H2 1be the Lorentzian hyperbolic plane and let (T∗Σ, gD)be the Riemannian extension of its Levi-Civita connection. Let JΣbe the para-complex structure on (Σ, gΣ)determined by the Lorentzian volume form and Ω−=−dιJΣ. Then it determines a para-Kähler structure (gD, J−)by gD(J−ξ, η) = Ω−(ξ, η) which is locally conformally flat with curvature tensor modelled on (M−), and thus locally isometric to the one in Theorem 2.1-(i). Furthermore, let (x1, x2)be a system of local coordinates on Σso that the metric tensor gΣ=1 (x1)2(dx1⊗dx1−dx2⊗dx2)and JΣ∂x1=∂x2,JΣ∂x2=∂x1. Let (x1, x2, x1′, x2′)be the induced coordinates on T∗Σ. It now follows that the paracomplex structure J−is given by J−∂x1′=−∂x2′, J−∂x1=∂x2−2x2′ x1∂x1′−2x1′ x1∂x2′, J−∂x2′=−∂x1′, J−∂x2=∂x1−2x1′ x1∂x1′−2x2′ x1∂x2′, with the corresponding Kähler two-form Ω−=−dιJΣ=dx1∧dx2′+dx2∧dx1′. 63 Locally conformally flat four-dimensional structures 2.3 Locally conformally flat four-dimensional null-Kähler structures In this section we will focus on those (1,1)-tensor fields that are two-step nilpotent, i.e., such that J2= 0. Such a tensor field is commonly said to be an almost tangent structure on M. The study of this kind of structures started with the works [51] and [52] around 1960 and they were further investigated during that decade in works such as [63] and [86]. If rank(J) = n,Jis said to be a tangent structure or, as we will call it in what follows, a null-Kähler structure (see [60,61]). Lemma 2.21. A null-Kähler surface (M, g, J0)is a Walker manifold. Proof. D= ker J= ImJis a Walker distribution on (M, g, J0). Indeed, since g(J0X, Y ) + g(X, J0Y)=0, then 0 = g(J0X, J0Y) + g(X, J2 0Y) = g(J0X, JY ), so Dis degenerate. Besides, since J0is parallel with respect to the Levi-Civita connection of g, ∇ZJ0X=J0∇ZX for all vector field Zon Mand all X∈ D, which shows that Dis also parallel. As a consequence from Lemma 2.21, we can consider the induced Walker coordinates (x1, x2, x1′, x2′)on M, with respect to which the metric is given by (2.1) and the Walker distribution is D= span{∂x1′, ∂x2′}. Since D= Im(J0)we can write J0∂x1=a∂x1′+b∂x2′and J0∂x2=c∂x1′+d∂x2′for some real-valued smooth functions a, b, c and don M. Imposing the condition for J0to be anti-self-adjoint we see that 0 = g(J0∂x1, ∂x1)+g(∂x1, J0∂x1)=2g(J0∂x1, ∂x1)=2g(a∂x1′+b∂x2′, ∂x1) = 2a and, analogously 2d= 0, c =−b. Hence, in Walker coordinates, the null-Kähler structure takes the form J0=λ(x1, x2, x1′, x2′)dx2′⊗∂ ∂x1−dx1′⊗∂ ∂x2. In this situation the associated two-form given by Ω(X, Y ) = g(JX, Y )takes the form Ω = λ(x1, x2, x1′, x2′)dx1∧dx2and a standard calculation shows that it is closed if and only if λ(x1, x2, x1′, x2′) = λ(x1, x2). 64 2.3 Locally conformally flat four-dimensional null-Kähler structures Theorem 2.22. Let (M, g, J)be a locally conformally flat null-Kähler structure. Then it is locally isometric to the cotangent bundle T∗Σof a strongly projectively flat affine surface (Σ, D)endowed with the Riemannian extension g=gD. Proof. The scalar curvature of a four-dimensional locally conformally flat Walker manifold τ= 3 tr(T) + 12ιX is necessarily zero, so the vector field Xmust vanish identically and the tensor field Tmust be trace-free. In this situation, the non-zero components of the covariant derivative ∇J0are given by (∇∂x1J0)∂x1=T11λx1′+T12λx2′−λ(Γ111+ Γ122) + ∂x1λ =−(∇∂x2J0)∂x1 (∇∂x1J0)∂x2=T21λx1′−T11λx2′−λ(Γ121+ Γ222) + ∂x2λ =−(∇∂x2J0)∂x2 (2.6) which are polynomials on the fibre coordinates. After analysing the conditions under which the expressions above vanish we easily deduce that the (1,1)-tensor field T must be zero. This implies that the metric must be a deformed Riemannian extension g=gD+π∗Φ, which, in these coordinates, takes the form g={−2Γijkyk+ Φij}dxi⊗dxj+dxi⊗dxi′+dxi′⊗dxi.(2.7) If we now consider ¯xi=xiand ¯xi′=xi′+ηifor some real-valued functions ηion the base surface, and write ∂¯xi=aik∂xk+bik∂xk′and ∂¯xi′=cik∂xk+dik∂xk′, we have 0 = d¯xj(∂¯xi′) = dxj(cik∂xk+dik∂xk′) = cij, δij=d¯xj′(∂¯xi′) = dxj′(dik∂xk′) = dij, δij=d¯xj(∂¯xi) = dxj(aik∂xk) = aij, 0 = d¯xj′(∂¯xi) = (dxj′+∂xℓηjdxℓ)(aik∂xk+bik∂xk′) = ∂xiηj+bij. so the coordinate vector fields are given by ∂¯xi=∂xi−∂xiηk∂ykand ∂¯xi′=∂xi′for some real-valued functions ηi,i= 1,2, on the base surface. It is easy to see that in these coordinates the metric takes the form g={−2Γijkxk′+2Γijkηk−∂xiηj−∂xjηi+Φij}dxi⊗dxj+dxi⊗dxi′+dxi′⊗dxi. Afifi showed in [1] that it is possible to find two smooth functions η1and η2on Σ such that Φij =−2Γijkηk+∂xiηj+∂xjηi, 65 Locally conformally flat four-dimensional structures or, equivalently, a one-form ηon Σsuch that Φ(∂xi, ∂xj)=(∇∂xiη)∂xj+ (∇∂xjη)∂xi.(2.8) Therefore, the (0,2)-tensor field Φcan be transformed away from Expression (2.7). At this point, the only conditions we have left for J0to be parallel are λ(Γ111+ Γ122) + ∂x1λ= 0, λ(Γ121+ Γ222) + ∂x2λ= 0,(2.9) and the compatibility condition for this system of partial differential equations reduces to λ∂x2(Γ122+ Γ111)−∂x1(Γ121+ Γ222)= 0.(2.10) Besides, it is known [62, Sec. 34] that if a Riemannian extension of the form g=gD+π∗Φis locally conformally flat, then the base affine surface (Σ, D)must be projectively flat. This means that there exists a one-form ωon Σsuch that Γijk=−(ωiδjk+ωjδik). This implies that the compatibility condition (2.10) can be written as −3∂x2ω(∂x1)+3∂x1ω(∂x2) = 0,(2.11) which means that the one-form ωmust be closed and therefore there exists a realvalued function flocally defined on Σsuch that ω=df. Consequently, (Σ, D) must be locally strongly projectively flat. Under these conditions the metric is locally conformally flat and null-Kähler. Conversely, if (Σ, D)is a strongly projectively flat affine surface, then there exist a one-form ωand a real-valued function fsuch that ω=df and the compatibility condition (2.11) automatically holds. 66 Chapter 3 Four-dimensional Kähler and para-Kähler Lie groups In this chapter we will describe all the left-invariant para-Kähler structures on fourdimensional Lie groups and analyse their geometry, thus completing the analysis previously carried out in [31, 32, 101]. The geometry of the left-invariant Kähler structures obtained by Ovando in [116] will also be clarified. The results in this chapter are contained in the work [70]. 3.1 Summary of results A para-Kähler Lie algebra is a triple (g, J, ⟨·,·⟩)so that J2= Id,⟨Jx, Jy⟩=−⟨x, y⟩,∇J= 0, for all vectors x, y ∈g. The associated two-form Ω(x, y) = ⟨Jx, y⟩is non-degenerate and closed, so (g,Ω) is a symplectic Lie algebra satisfying Ω(Jx, Jy) = −Ω(x, y). The eigenspaces L= ker (J−Id) and L′= ker (J+ Id) are Lagrangian subalgebras and g=L⊕L′ is a Lagrangian decomposition of (g,Ω). For a fixed symplectic structure Ωon a Lie algebra gwe will describe all the para-Kähler structures (J, ⟨·,·⟩,Ω), up to isometric automorphisms preserving the symplectic structure and modulo reversing the metric – in both cases the corresponding Lagrangian decomposition g=L⊕L′is preserved. The different Lagrangian decompositions, which are of much interest in the para-Kähler setting, will also be explicitly described in each case. The geometry of four-dimensional para-Kähler Lie groups naturally splits into the symmetric and the non-symmetric cases. The latter splits into the semi-symmetric and the non-semi-symmetric situations. The symmetric case can be summarized as follows. 67 Four-dimensional Kähler and para-Kähler Lie groups Theorem 3.1. Let (G, ⟨·,·⟩, J)be a non-flat locally symmetric four-dimensional para-Kähler Lie group. Then, there are two distinct situations. (i) If the Ricci operator is diagonalizable, then one of the following holds: (i.a) The Ricci operator vanishes and the anti-self-dual Weyl curvature operator is two-step nilpotent. (i.b) The para-holomorphic sectional curvature is a non-zero constant. (i.c) The metric is Einstein with non-zero scalar curvature, and the self-dual and anti-self-dual Weyl curvature operators are diagonalizable with the same eigenvalues. (i.d) The manifold is locally a product of two surfaces of constant Gaussian curvature. The self-dual and anti-self-dual Weyl curvature operators are diagonalizable with the same eigenvalues. (ii) If the Ricci operator is non-diagonalizable, then one of the following holds: (ii.a) The Ricci operator has complex eigenvalues. The self-dual and anti-selfdual Weyl curvature operators are diagonalizable with the same eigenvalues. (ii.b) The Ricci operator is two-step nilpotent and the anti-self-dual Weyl curvature operator either vanishes or is two-step nilpotent. The structures in Assertion (i.a), which do not have a Kählerian counterpart, are realized on r4,−1,r4,−1,−1and d4,1, and they all correspond to symmetric Osserman manifolds – which are four-dimensional Einstein manifolds that are either self-dual or anti-self-dual – with non-diagonalizable Jacobi operators [77]. A para-Kähler manifold (M, g, J)is said to be opposite para-Kähler if there exists a para-Kähler structure (J′, g)so that JJ′=J′J. In the four-dimensional setting, the corresponding para-Kähler forms Ωand Ω′induce opposite orientations and Q=JJ′defines a parallel product structure on the manifold. In this situation, (M, g)locally splits as a product of two oriented surfaces M=N1×N2so that J=J1⊕J2and J′=J1⊕ −J2, where Jiis the para-complex structure induced by the volume form on Ni, for i= 1,2. Conversely, a four-dimensional product of two oriented Lorentzian surfaces naturally inherits a para-Kähler and opposite paraKähler structure. This is the case of the structures corresponding to Assertion (i.d) in Theorem 3.1. As an immediate consequence of Theorem 3.1 and the results of Chapter 2 we conclude that locally conformally flat left-invariant para-Kähler structures can be modelled as follows. 68 3.3 Para-Kähler structures on r4,0 it is easy to see that any symplectic form on r4,0is symplectomorphically equivalent to Ωε=e14 +εe23, ε2= 1. Now, the automorphisms preserving the symplectic structure (r4,0,Ωε)are those determined by Φabove satisfying the conditions Φ1:z11 = 1, z24 = 0, z34 = 0, z22 = 1, Φ2:z11 = 1, z24 = 0, z34 = 0, z22 =−1. Bearing this in mind, we determine the conditions that are necessary for the compatibility of the para-complex structure with the symplectic structures and the integrability of the para-complex structure (i.e., the integrability of the eigenspaces ker(J∓Id) corresponding to the eigenvalues ±1of J, which is equivalent to the vanishing of the Nijenhuis tensor). The compatibility condition Ωε(J·, J·) = −Ωε(·,·)determines the following system of polynomial equation on the components {aij} εa31 −a42 = 0,−(εa21 +a43) = 0,−ε(a22 +a33) = 0, −(a11 +a44) = 0,−(a12 +εa34) = 0,−a13 +εa24 = 0, and so the para-complex structure Jmust be such that Je1=a11e1+a21e2+a31e3+a41e4, Je3=εa24e1+a23e2−a22e3−εa21e4, Je2=−εa34e1+a22e2+a32e3+εa31e4, Je4=a14e1+a24e2+a34e3−a11e4. Now, a straightforward calculation shows that the non-zero components of the Nijenhuis tensor are determined by NJ(e2, e1) = εa34a41e1−(a32a41 +εa31(a21 −a31)) e2−εa2 31e3−εa31a41e4, NJ(e3, e1) = −εa41(a24 +a34)e1+ (2a22a41 +εa21(a21 −a31)) e2 + (a32a41 +εa21a31)e3+εa41(a21 +a31)e4, NJ(e3, e2) = −(a24a31 −(a21 −a31)a34)e1+ε(2a22a31 −a21a32)e2 +εa31a32e3+a2 31e4, NJ(e4, e1) = 1−a2 11 −a14a41 +εa31a34e1−(a31(a11 +a22) + a34a41)e2 −a31a32e3−εa2 31e4, 75 Four-dimensional Kähler and para-Kähler Lie groups NJ(e4, e2) = −ε(a14a31−(2a11 +a32)a34)e1 −(a32(a11 +a22)−ε(a21 −a31)a34)e2+ (εa31a34 −a2 32)e3 +ε(a34a41 −a31a32)e4, NJ(e4, e3) = ε(a14a21 −a22a34 −a11(2a24 +a34)) e1 + (1 + a22(2a11 +a22) + εa21(a34 −a24)) e2 + (a32(a11 +a22)−εa24a31)e3+ε(a31(a11 +a22)−a24a41)e4, which again determine a system of polynomial equations on {aij}that needs to be solved in order to obtain the integrability conditions on J. It immediately follows from the expressions of NJ(e2, e1)and NJ(e4, e2)that both a31 and a32 must vanish. In this situation NJ(e3, e2) = a21a34e1 and so there are two different possibilities depending on whether a34 = 0 or not. A straightforward calculation shows that there can be no para-Kähler structures with a34 = 0. Therefore, we can assume that a34 = 0. Then the condition J2= Id gives the following system of polynomial equations a2 11 +a14a41 = 0, a21(a11 +a22) + a24a41 = 0, a2 22 −1 = 0, a14a21 +a24(a22 −a11) = 0, from where it follows that a22 =ε2, with ε2 2= 1. At this point, NJ(e3, e1) = −εa24a41e1+ (2ε2a41 +εa2 21)e2+εa21a41e4, which implies that there are again two different possibilities depending on whether or not a41 = 0. A straightforward calculation shows that the condition a41 = 0 is incompatible with the integrability of the para-complex structure. Consequently, we assume that a41 = 0. Now, NJ(e3, e1) = εa2 21e2, NJ(e4, e1) = 1−a2 11e1, so a21 = 0 and a11 =ε1, for ε2 1= 1. At this point, the only non-zero component of the Nijenhuis tensor is NJ(e4, e3) = −2ε1εa24e1+ 2(1 + ε1ε2)e2. Therefore, a24 = 0 and ε2=−ε1, and the condition J2= Id automatically holds. What we have just obtained is the para-Kähler structures (J, ⟨·,·⟩)on r4,0given by    Je1=ε1e1, Je2=−ε1e2, Je3=a23e2+ε1e3, Je4=a14e1−ε1e4, ⟨·,·⟩ = 2(−ε1e1◦e4−ε1ε e2◦e3)−εa23e3◦e3−a14e4◦e4. 76 3.4 Para-Kähler structures on r4,−1 It is easy to see that all these structures are symplectomorphically equivalent to (J, ⟨·,·⟩) :    Je1=−e1, Je2=e2, Je3=−e3, Je4=e4, ⟨·,·⟩ = 2(e1◦e4−ε e2◦e3) through the symplectomorphism determined by Φ1satisfying z14 =−ε1 2a14 and z23 =ε1 2a23. These structures correspond to the decomposition of (r4,0,Ωε)as a direct sum of Lagrangian subalgebras as r4,0=L⊕L′= span{e2, e4}⊕span{e1, e3}, where Land L′are the eigenspaces associated to the eigenvalues ±1of the paracomplex structure J, respectively. The structures (r4,0, J, ⟨·,·⟩)are Ricci-flat and a straightforward calculation shows that their curvature tensors satisfy ∇R= 2e4⊗R, so they are recurrent with recurrence one-form ξ= 2e4. Besides, the corresponding curvature tensors acting on the space of two-forms are given by R(e3∧e4) = −e1∧e2=εR(e3, e4, e3, e4)e1∧e2. Therefore, the para-Kähler structures (r4,0, J, ⟨·,·⟩)are special recurrent and, consequently, simply harmonic, so they are locally modelled on (3.1) and their curvature tensors are semi-symmetric. These structures are thus covered by Theorem 3.6. Finally, direct calculations similar to the ones described above show that the structures (J, ⟨·,·⟩)admit opposite almost para-Kähler structures compatible with the opposite symplectic forms Ω′=e14 −εe23 −µe34, with µ∈R. 3.4 Para-Kähler structures on r4,−1 Let {e1, e2, e3, e4}be a basis of the Lie algebra r4,−1determined by [e1, e4] = −e1,[e2, e4] = e2,[e3, e4] = e3−e2, which, according to [6], corresponds to the Lie algebra of the semi-direct extension Re4⋉φR3where φ= ad(e4): φ(e1) = e1, φ(e2) = −e2, φ(e3) = e3−e2. Any symplectic form on r4,−1is of the form ω=α13e13 +α14e14 +α24e24 +α34e34, α13α24 = 0, 77 Four-dimensional Kähler and para-Kähler Lie groups and the automorphisms of this Lie algebra are given by Φ =        z11 0 0 z14 0z22 z23 z24 0 0 z22 z34 0 0 0 1        with z11z22 = 0. Now, it is not difficult to check that any symplectic structure is symplectomorphically equivalent to Ω = e13 +e24.Moreover, the automorphisms preserving the symplectic structure (r4,−1,Ω) are given by Φabove with z11 = 1,z22 = 1,z34 = 0 and z23 =z14. Now, a straightforward calculation as in the previous section shows that any para-Kähler structure on r4,−1is equivalent to (J, ⟨·,·⟩) :    Je1=−e1, Je2=e2, Je3=−κe1+e3, Je4=−e4, ⟨·,·⟩ = 2(e1◦e3−e2◦e4) + κ e3◦e3, κ ∈R, which correspond to the Lagrangian decomposition r4,−1=L⊕L′= span{e2, e3−κ 2e1}⊕span{e1, e4}. The structures above are flat if κ= 0. Otherwise, they are Ricci-flat, locally symmetric – thus covered by Theorem 3.1 – and the underlying structures are locally modelled on (3.1) with Ψ(x1, x2) = ±(x2)2. Furthermore, the structures above admit opposite almost para-Kähler structures which are compatible with the opposite symplectic form Ω′=e13 −e24 +µe34 with µ∈R. 3.5 Para-Kähler structures on r2r2 Let {e1, e2, e3, e4}be a basis of the Lie algebra r2r2determined by [e1, e2] = e2,[e3, e4] = e4, which, according to [6], corresponds to the Lie algebra of the non-unimodular semidirect extension E(1,1) ⋊ R isomorphic to aff(R)×aff(R)as in Lemma 1.9. 78 3.5 Para-Kähler structures on r2r2 We make use of Lie algebra automorphisms to simplify the final expressions of the structures. The automorphisms of this Lie algebra are given by Φ1=        0 0 1 0 0 0 z23 z24 1 0 0 0 z41 z42 0 0        , z24z42 = 0 Φ2=        1 0 0 0 z21 z22 0 0 0 0 1 0 0 0 z43 z44        . z22z44 = 0 Any symplectic structure on r2r2is given by ω=α12e12 +α13e13 +α34e34, α12α34 = 0, and Ovando showed in [115] that these (r2r2, ω)are symplectomorphically equivalent to Ωλ=e12 +e34 +λe13 with λ≥0. It is easy to check that the symplectic form Ωλis preserved by any automorphism Φ2 with z22 =z44 = 1 (in the special case where λ= 0, the automorphisms Φ1with z24 =z42 = 1 also preserve the symplectic form Ω0). Considering the action of the symplectomorphisms Φ2that preserve Ωλwe are allowed to restrict our study to the para-complex structures J= (aij)satisfying one of the following conditions. Case 1. a12 = 0 and a14 = 0. Case 2. a12 = 0 and a14 = 0, in which case one may also assume a13 = 0. Case 3. a12 = 0 and a14 = 0, in which case one may also assume a11 = 0. Case 4. a12 = 0 and a14 = 0, in which case one may also assume a11 =a13 = 0. In each case we determine the necessary conditions for the compatibility of the paracomplex structure with the symplectic structures and the integrability of the latter. To do so, we distinguish the two possibilities corresponding to the symplectic structures Ωλ(with λ= 0) and Ω0, since they give rise to different geometries. 3.5.1 Para-Kähler structures on (r2r2,Ωλ)with λ > 0 A straightforward calculation reveals that the Cases 2, 3 and 4 above are not compatible with the existence of para-Kähler structures. Therefore we focus on the case a12 =a14 = 0. Long but straightforward calculations show that the para-Kähler structures (J, ⟨·,·⟩)which are compatible with Ωλare symplectomorphically equivalent to one of the following three families of para-Kähler structures (J11,⟨·,·⟩11) :    J11e1=−e1, J11e2=e2, J11e3=e3, J11e4=−e4, ⟨·,·⟩11 = 2(e1◦e2+λe1◦e3−e3◦e4), 79 Four-dimensional Kähler and para-Kähler Lie groups (J12,⟨·,·⟩12) :            J12e1=−e1+ 2λe2−2e3, J12e2=e2, J12e3=e3, J12e4= 2e2−e4, ⟨·,·⟩12 = 2(e1◦e2+λe1◦e3+ 2e1◦e4−e3◦e4), (J13,⟨·,·⟩13) :            J13e1=−e1, J13e2=e2−2e4, J13e3= 2e1+e3+ 2λe4, J13e4=−e4, ⟨·,·⟩13 = 2(e1◦e2+λe1◦e3−2e2◦e3−e3◦e4). The structures above correspond to the decompositions r2r2=L1i⊕L′ 1ias direct sums of Lagrangian subalgebras given by r2r2=L11 ⊕L′ 11 = span{e2, e3}⊕span{e1, e4} =L12 ⊕L′ 12 = span{e2, e3}⊕span{e4−e2, e1+e3−λe2} =L13 ⊕L′ 13 = span{e4−e2, e1+e3+λe2}⊕span{e1, e4}. Moreover, a straightforward calculation shows that all the structures above are non-flat and Ricci-flat with recurrent curvature, i.e., their curvature tensors satisfy ∇1iR1i=ξ1i⊗R1i, and their recurrence one-forms ξ1iare given by ξ11 = 2(e1+e3), ξ12 = 2e3, ξ13 = 2e1. Furthermore, the corresponding curvature operators R1i: Λ2→Λ2acting on the space of two-forms are given by R1i(e1∧e2) = −λe2∧e4=R1i(e1, e3, e3, e1)e2∧e4,for i= 1,2,3, from where it follows that all the para-Kähler structures above are special recurrent and thus locally modelled on (3.1). Consequently, their curvature tensors are semisymmetric and they are covered by Theorem 3.6. Finally, the structures (J11,⟨·,·⟩11)admit opposite almost para-Kähler structures compatible with the opposite symplectic forms Ω′ 11 =e12 −e34 + (µ−λ)e13 with µ∈R, while the structures (J12,⟨·,·⟩12)and (J13,⟨·,·⟩13)do not admit opposite almost para-Kähler structures. 80 3.5 Para-Kähler structures on r2r2 3.5.2 Para-Kähler structures on (r2r2,Ω0) Proceeding as in the previous case, it is straightforward to check that no para-Kähler structures exist in Case 2 while the other three cases give rise to three essentially different geometries. Para-Kähler structures of constant para-holomorphic sectional curvature Assuming that a12 = 0 and a14 = 0 as in Case 4, any para-Kähler structure is equivalent to the structures (J21,⟨·,·⟩21), which are given by          J21e1=e3−1 κe2, J21e2=−κ(e1+e3), J21e4=e4−e2−κ(e1+e3), J21e3=−e3, ⟨·,·⟩21 =κ(e2◦e2+e4◦e4+ 2e2◦e4)−2e1◦e4+ 2e3◦e4−1 κe1◦e1, where κ= 0. These correspond to the Lagrangian decompositions r2r2=L21 ⊕L′ 21 = span{e4−e2, e1+e3−1 κe2}⊕span{e3, e2+κe1}. Besides, one can easily check that their para-holomorphic sectional curvatures are constantly H21 =κ, so these structures correspond to Theorem 3.1-(i.b). Flat para-Kähler structures Assuming that a12 = 0 and a14 = 0 as in Case 1, if a34 = 0, then there are three inequivalent flat para-Kähler structures given by (J22,⟨·,·⟩22) :          J22e1=−e1, J22e2=e2−2e4, J22e3= 2e1+e3, J22e4=−e4, ⟨·,·⟩22 = 2(e1◦e2−2e2◦e3−e3◦e4), (J23,⟨·,·⟩23) :    J23e1=−e1, J23e2=e2, J23e3=εe3, J23e4=−εe4, ⟨·,·⟩23 = 2(e1◦e2−εe3◦e4), ε =±1, (J24,⟨·,·⟩24) :          J24e1=−e1−2e3, J24e2=e2, J24e3=e3, J24e4= 2e2−e4, ⟨·,·⟩24 = 2(e1◦e2+ 2e1◦e4−e3◦e4), 81 Four-dimensional Kähler and para-Kähler Lie groups which correspond to the Lagrangian decompositions r2r2=L2i⊕L′ 2igiven by r2r2=L22 ⊕L′ 22 = span{e1+e3, e4−e2}⊕span{e1, e4} =L23 ⊕L′ 23 = span{e2, e3}⊕span{e1, e4}(ε= 1) =L23 ⊕L′ 23 = span{e2, e4}⊕span{e1, e3}(ε=−1) =L24 ⊕L′ 24 = span{e2, e3}⊕span{e1+e3, e4−e2}. Para-Kähler and opposite para-Kähler structures The remaining possibilities corresponding to Case 1 with a34 = 0 and Case 3 give rise to two families of para-Kähler structures determined by (J25,⟨·,·⟩25) :    J25e1=−1 κ1e2, J25e2=−κ1e1, J25e3=−e3, J25e4=e4, ⟨·,·⟩25 =−1 κ1e1◦e1+κ1e2◦e2+ 2e3◦e4, (J26,⟨·,·⟩26) :          J26e1=−1 κ1e2, J26e2=−κ1e1, J26e3=−1 κ2e4, J26e4=−κ2e3, ⟨·,·⟩26 =−1 κ1e1◦e1+κ1e2◦e2−1 κ2e3◦e3+κ2e4◦e4, where κ1κ2= 0. The corresponding Lagrangian decompositions are given by r2r2=L25 ⊕L′ 25 = span{e4, e2−κ1e1}⊕span{e3, e2+κ1e1} =L26 ⊕L′ 26 = span{e4−κ2e3, e2−κ1e1}⊕span{e4+κ2e3, e2+κ1e1}. Now, it is straightforward to see that both metrics are locally symmetric, so they are covered by Theorem 3.1-(i.d). Their Ricci operators are diagonalizable with respect to the basis {e1, e2, e3, e4}with Ricci curvatures Ric25 = diag[κ1, κ1,0,0] and Ric26 = diag[κ1, κ1, κ2, κ2]. Therefore, the underlying pseudo-Riemannian manifolds split locally as a product of two Lorentzian surfaces of constant Gaussian curvature, namely, N1(κ1)×L2 or N1(κ1)×N2(κ2), where L2denotes the Minkowskian plane. Thus they also admit opposite para-Kähler structures. Furthermore, the metrics ⟨·,·⟩26 are Einstein if κ1=κ2and locally conformally flat if κ1=−κ2, in which case they correspond to the metrics in Corollary 3.2-(iii). 82 3.6 Para-Kähler structures on rh3 3.6 Para-Kähler structures on rh3 Let {e1, e2, e3, e4}be a basis of the Lie algebra rh3determined by [e1, e2] = e3, which corresponds to the Lie algebra of either H3×Ror the semi-direct extension Re1⋉φR3given by φ= ad(e1): φ(e2) = e3, φ(e3) = 0, φ(e4) = 0. Symplectic forms on rh3are given by ω=α12e12 +α13e13 +α14e14 +α23e23 +α24e24, α14α23 −α13α24 = 0, as shown in [115]. All these symplectic structures (rh3, ω)are symplectomorphically equivalent to Ω = e14 +e23 through an automorphism of the form Φ =        z11 z12 0 0 z21 z22 0 0 z31 z32 z11z22 −z12z21 z34 z41 z42 0z44        ,with (z11z22 −z12z21)z44 = 0. The ones that preserve the symplectic structure (rh3,Ω) determined by z11z2 22 = 1, z12z22 =−z34,z21 = 0,z42 =z3 22z31 −z22z34z41 and z44 =z2 22. A long but straightforward calculation shows that any para-Kähler structure on rh3must be flat and equivalent to one of the following two para-Kähler structures (J1,⟨·,·⟩1) :    J1e1=e1, J1e2=−e2, J1e3=e3, J1e4=−e4, ⟨·,·⟩1= 2(−e1◦e4+e2◦e3), (J2,⟨·,·⟩2) :    J2e1=−e2, J2e2=−e1, J2e3=e4, J2e4=e3, ⟨·,·⟩2= 2(e1◦e3+e2◦e4). In each case, rh3decomposes as a direct sum of Lagrangian subalgebras as rh3=L1⊕L′ 1= span{e1, e3}⊕span{e2, e4} =L2⊕L′ 2= span{e3+e4, e2−e1}⊕span{e1+e2, e4−e3}. 83 Four-dimensional Kähler and para-Kähler Lie groups 3.7 Para-Kähler structures on rr3,0 Let {e1, e2, e3, e4}be a basis of the Lie algebra rr3,0determined by [e1, e2] = e2, which corresponds to the Lie algebra of E(1,1) ×Ror to the semi-direct extension Re1⋉φR3given by φ= ad(e1): φ(e2) = e2, φ(e3) = 0, φ(e4) = 0. The symplectic structures on rr3,0are given by ω=α12e12 +α13e13 +α14e14 +α34e34, α12α34 = 0, as shown in [115]. All these symplectic structures (rr3,0, ω)are symplectomorphically equivalent to Ω = e12 +e34 through an automorphism of the form Φ =        1 0 0 0 z21 z22 0 0 z31 0z33 z34 z41 0z43 z44        ,with (z33z44 −z34z43)z22 = 0. Moreover, the automorphisms preserving the symplectic structure (rr3,0,Ω) are the ones determined by Φabove satisfying Φ1:z22 = 1, z31 = 0, z41 = 0, z44z33 = 1 + z34z43 with z33 = 0, Φ2:z22 = 1, z31 = 0, z41 = 0, z33 = 0, z43z34 =−1. Proceeding as in the previous cases, a straightforward calculation shows that any para-Kähler structure on (rr3,0,Ω) is equivalent to one of the following two families (J1,⟨·,·⟩1) :    J1e1=ε e1, J1e2=−ε e2, J1e3=e3, J1e4=−e4, ⟨·,·⟩1=−2(ε e1◦e2+e3◦e4), ε =±1, (J2,⟨·,·⟩2) :    J2e1=1 κe2, J2e2=κe1, J2e3=e3, J2e4=−e4, ⟨·,·⟩2=1 κe1◦e1−κe2◦e2−2e3◦e4, κ = 0. 84 3.11 Para-Kähler structures on r4,−1,−1 Para-Kähler and opposite para-Kähler structures on r4,−1,β If a43 = 0, then any para-Kähler structure is equivalent to (J3,⟨·,·⟩3) :    J3e1=−e1, J3e2=e2, J3e3=κe4, J3e4=1 κe3, ⟨·,·⟩3= 2e1◦e2+κe3◦e3−1 κe4◦e4, κ = 0, so that r4,−1,β =L3⊕L′ 3= span{e2, e4+1 κe3}⊕span{e1, e4−1 κe3}. The metrics above are locally symmetric with diagonalizable Ricci operator Ric3= diag[0,0, κβ2, κβ2]. Therefore, the underlying manifolds are locally isometric to a product L2×Nof the Minkowskian plane and a Lorentzian surface of constant sectional curvature KN=κβ2. Consequently, they are para-Kähler and opposite para-Kähler and they correspond to the metrics in Theorem 3.1-(i.d). 3.11 Para-Kähler structures on r4,−1,−1 Let {e1, e2, e3, e4}be a basis of the Lie algebra r4,−1,−1determined by [e1, e4] = −e1,[e2, e4] = e2,[e3, e4] = e3. This Lie algebra corresponds to the semi-direct extension Re4⋉φR3given by φ= ad(e4): φ(e1) = e1, φ(e2) = −e2, φ(e3) = −e3. Any symplectic structure on r4,−1,−1is of the form ω=α12e12 +α13e13 +α14e14 +α24e24 +α34e34, α13α24 −α12α34 = 0, and all of them are symplectomorphically equivalent to Ω = e12 +e34 through an automorphism of r4,−1,−1, which are given by Φ =        z11 0 0 z14 0z22 z23 z24 0z32 z33 z34 0 0 0 1        with z11(z22z33 −z23z32)= 0. Moreover, the automorphisms preserving the symplectic structure (r4,−1,−1,Ω) are the ones above with z23 =z24 = 0,z33 = 1,z32 =z14z22 and z22z11 = 1. The associated para-Kähler structures split into two cases depending on whether a43 = 0 or a43 = 0 as follows. 91 Four-dimensional Kähler and para-Kähler Lie groups Para-Kähler and opposite para-Kähler structures on r4,−1,−1 Straightforward calculations as in the previous sections show that any para-Kähler structure with a43 = 0 is equivalent to (J1,⟨·,·⟩1) :    J1e1=−e1, J1e2=e2, J1e3=κe4, J1e4=1 κe3, ⟨·,·⟩1= 2e1◦e2+κe3◦e3−1 κe4◦e4, κ = 0, so that r4,−1,−1=L1⊕L′ 1= span{1 κe3+e4, e2}⊕span{e1, e4−1 κe3}. It is not difficult to check that these structures are locally symmetric and their Ricci operators are diagonalizable Ric1= diag[0,0, κ, κ]. Therefore, the underlying pseudo-Riemannian structures split locally as a product L2×Nof a Lorentzian surface Nof constant Gaussian curvature KN=κand the Minkowskian plane. As a consequence, they also admit opposite para-Kähler structures. These structures correspond to those in Theorem 3.1-(i.d). Ricci-flat para-Kähler structures on r4,−1,−1 Assuming that a43 = 0, the para-Kähler structures on (r4,−1,−1,Ω) are equivalent to one of the following families, which correspond to different geometric situations. (J2,⟨·,·⟩2) :    J2e1=−e1, J2e2=−κe1+e2, J2e3=e3, J2e4=−e4, ⟨·,·⟩2=κe2◦e2+ 2(e1◦e2−e3◦e4), κ = 0,±1, so that r4,−1,−1=L2⊕L′ 2= span{e3, e2−κ 2e1}⊕span{e4, e1}, (J3,⟨·,·⟩3) :    J3e1=−e1+κe2, J3e2=e2, J3e3=−e3, J3e4=e4, ⟨·,·⟩3=κe1◦e1+ 2(e1◦e2+e3◦e4), κ = 0,±1, so that the Lie algebra decomposes as r4,−1,−1=L3⊕L′ 3= span{e2, e4}⊕span{e3, e2−2 κe1}(κ= 0) = span{e2, e4}⊕span{e3, e1}(κ= 0), or (J4,⟨·,·⟩4) :    J4e1=κe2+e4, J4e2=e3, J4e3=e2, J4e4=e1−κe3, ⟨·,·⟩4=κ(e1◦e1+e4◦e4) + 2(e1◦e3−e2◦e4), κ ∈R, 92 3.11 Para-Kähler structures on r4,−1,−1 so that (r4,−1,−1,Ω) decomposes as direct sums of Lagrangian subalgebras r4,−1,−1=L4⊕L′ 4= span{e2+e3, e1+e4+κe2}⊕span{e3−e2, e4−e1+κe2}. The structures (J2,⟨·,·⟩2)are flat if κ= 0. Otherwise, they are Ricci-flat and locally symmetric, thus locally modelled on (3.1) with Ψ(x1, x2) = ±(x2)2. Moreover, these structures admit opposite almost para-Kähler structures compatible with the opposite symplectic two-forms Ω′ 2=e12 −e34 +µe24 with µ∈R. These structures correspond to Theorem 3.1-(i.a). The structures (J3,⟨·,·⟩3)are flat if κ= 0 and Ricci-flat otherwise. They are special recurrent with recurrence one-form ξ3=−4e4and curvature operator determined by R3(e1∧e4) = −3κe2∧e3=R3(e1, e4, e1, e4)e2∧e3. Therefore, these structures are simply harmonic and locally modelled on (3.1), and their curvature tensor is semi-symmetric. Furthermore, they admit opposite almost para-Kähler structures compatible with the opposite symplectic two-forms Ω′ 3=e12 −e34 +µe14, µ ∈R. What is more, they have an associated one-parameter family of hypersymplectic structures (J3, Jδ,⟨·,·⟩3)given by the Kähler structures Jδe1=−κδ 2e3−1 δe4, Jδe2=−δe3, Jδe3=1 δe2, Jδe4=δe1−κδ 2e2, so that JδJ3=−J3Jδfor any δ= 0 (see Remark 3.4 and [5] for more information). Finally, the structures (J4,⟨·,·⟩4)are flat if κ= 0. Otherwise, they are Ricci-flat and special recurrent with recurrence one-form ξ4=−4e4and curvature operators given by R4(e1∧e4) = −3κe2∧e3=R4(e1, e4, e1, e4)e2∧e3. Therefore, they are simply harmonic and locally modelled on (3.1), so their curvature tensors are semi-symmetric. Moreover, these structures admit opposite almost paraKähler structures compatible with the opposite symplectic two-forms Ω′ 4=e13 +e24 +µe14, µ ∈R. The structures (J3,⟨·,·⟩3)and (J4,⟨·,·⟩4)are covered by Theorem 3.6. 93 Four-dimensional Kähler and para-Kähler Lie groups 3.12 Para-Kähler structures on r4,−α,α Let {e1, e2, e3, e4}be a basis of the Lie algebra r4,−α,α determined by [e1, e4] = −e1,[e2, e4] = αe2,[e3, e4] = −αe3, where 0< α < 1. According to [6], this Lie algebra corresponds to the semi-direct extension Re4⋉φR3given by φ= ad(e4): φ(e1) = e1, φ(e2) = −αe2, φ(e3) = αe3. The symplectic forms on r4,−α,α are given by ω=α14e14 +α23e23 +α24e24 +α34e34, α14α23 = 0, and all of them are symplectomorphically equivalent to Ω = e14 +e23 through a Lie algebra automorphism of the form Φ =        z11 0 0 z14 0z22 0z24 0 0 z33 z34 0 0 0 1        ,with z11z22z33 = 0. Moreover, the automorphisms preserving the symplectic structure (r4,−α,α,Ω) are given by Φabove with z11 = 1,z24 =z34 = 0 and z33z22 = 1. The different classes of para-Kähler structures on this Lie algebra arise from the cases a41 = 0 and a41 = 0, which we study separately in what follows. Para-Kähler and opposite para-Kähler structures on r4,−α,α Straightforward calculations as in the previous sections show that any para-Kähler structure with a41 = 0 is equivalent to (J1,⟨·,·⟩1) :    J1e1=κe4, J1e2=−e2, J1e3=e3, J1e4=1 κe1, ⟨·,·⟩1=κe1◦e1−1 κe4◦e4+ 2e2◦e3, κ = 0, which induces the decompositions r4,−α,α =L1⊕L′ 1= span{e3, e4+1 κe1}⊕span{e2, e4−1 κe1} 94 3.12 Para-Kähler structures on r4,−α,α of r4,−α,α as direct sums of Lagrangian subalgebras. These structures are locally symmetric and their Ricci operators are diagonalizable Ric1= diag[0,0, κ, κ]. Therefore, the underlying manifolds split locally as products L2×Nof a Lorentzian surface Nof constant Gaussian curvature KN=κand the Minkowskian plane. Consequently, they also admit opposite para-Kähler structures and are covered by Theorem 3.1-(i.d). Ricci-flat para-Kähler structures Assuming that a41 = 0, the corresponding para-Kähler structures are equivalent to one of the following two families of para-Kähler structures. (J2,⟨·,·⟩2) :    J2e1=−e1, J2e2=−e2+κe3, J2e3=e3, J2e4=e4, ⟨·,·⟩2=κe2◦e2+ 2(e1◦e4+e2◦e3), κ = 0,±1, which induces the Lagrangian decompositions r4,−α,α =L2⊕L′ 2= span{e3, e4}⊕span{e1, e3−2 κe2}(κ= 0) = span{e3, e4}⊕span{e1, e2}(κ= 0), or (J3,⟨·,·⟩3) :    J3e1=e1, J3e2=−e2, J3e3=−κe2+e3, J3e4=−e4, ⟨·,·⟩3=κe3◦e3+ 2(e2◦e3−e1◦e4), κ = 0,±1, so that r4,−α,α =L3⊕L′ 3= span{e1, e3−κ 2e2}⊕span{e2, e4}. The structures (J2,⟨·,·⟩2)are flat if κ= 0, while structures (J3,⟨·,·⟩3)are flat if either κ= 0 or α=1 2. Otherwise, all the structures above are Ricci-flat and special recurrent with recurrence one-forms given by ξ2= 2(1 + α)e4and ξ3= 2(1 −α)e4, respectively, and curvature operators determined by R2(e2∧e4) = α(1 + 2α)κe1∧e3=R2(e2, e4, e4, e2)e1∧e3, R3(e3∧e4) = α(1 −2α)κe1∧e2=R3(e3, e4, e3, e4)e1∧e2. Therefore, they are simply harmonic and locally modelled on (3.1), so their curvature tensor is semi-symmetric. Furthermore, these structures admit opposite almost paraKähler structures compatible with the opposite symplectic forms given by Ω′ 2=e14 −e23 +µe24,Ω′ 3=e14 −e23 +µe34, where µ∈R. All these structures are covered by Theorem 3.6. 95 Four-dimensional Kähler and para-Kähler Lie groups 3.13 Para-Kähler structures on h4 Let h4be the Lie algebra generated by {e1, e2, e3, e4}, so that [e1, e2] = e3,[e1, e4] = −1 2e1,[e2, e4] = −e1−1 2e2,[e3, e4] = −e3. According to [6], this Lie algebra corresponds to the semi-direct extension Re4⋉φ H3given by φ= ad(e4): φ(e1) = 1 2e1, φ(e2) = e1+1 2e2, φ(e3) = e3. The symplectic structures on h4, which are given by ω=α12(e12 −e34) + α14e14 +α24e24, α12 = 0, are equivalent to Ωε=ε(e12 −e34),ε=±1, through a Lie algebra automorphism Φ =        z22 z12 0z14 0z22 0z24 2z22z24 2(z12 + 2z22)z24 −2z14z22 z2 22 z34 0 0 0 1        with z22 = 0. Moreover, the Lie algebra automorphisms preserving the symplectic structure (h4,Ωε) are given by Φabove with z24 =z14 = 0 and z22 =±1. Now, it is easy to see that any para-Kähler structure on (h4,Ωε)is equivalent to (J, ⟨·,·⟩) :    Je1=−ε e1, Je2=ε e2, Je3=ε e3, Je4=−ε e4, ⟨·,·⟩ = 2(e1◦e2+e3◦e4), ε =±1, which correspond to the Lagrangian decomposition h4=L⊕L′= span{e2, e3}⊕span{e1, e4}. A straightforward calculation shows that the structures (J, ⟨·,·⟩)are Ricci-flat with recurrent curvature. The recurrence one-form is given by ξ=e4and their curvature operator is determined by R(e2∧e4) = −e1∧e3=R(e2, e4, e2, e4)e1∧e3. As a consequence, (h4, J, ⟨·,·⟩)are simply harmonic manifolds with special recurrent curvature and locally modelled on (3.1) whose curvature tensor is semi-symmetric and so they are covered by Theorem 3.6. A straightforward calculation shows that the structures (h4, J, ⟨·,·⟩)do not admit any opposite almost para-Kähler structures. 96 3.14 Para-Kähler structures on d4,1 Remark 3.10.These structures, together with the structures (J13,⟨·,·⟩13)on (d4,1,Ω1 ) given in Section 3.14.1, (J1i,⟨·,·⟩1i),i= 2,3, on (d4,2,Ω1)given in Section 3.17.1, and (J32,⟨·,·⟩32)on (d4,2,Ω3)given in Section 3.17.3, are the only Ricci-flat paraKähler structures that do not admit any left-invariant opposite almost para-Kähler structures. 3.14 Para-Kähler structures on d4,1 Let d4,1be the Lie algebra generated by {e1, e2, e3, e4}so that [e1, e2] = e3,[e1, e4] = −e1,[e3, e4] = −e3. According to [6], this Lie algebra corresponds to the semi-direct extension Re4⋉φ H3given by φ= ad(e4): φ(e1) = e1, φ(e2) = 0, φ(e3) = e3. The symplectic structures on d4,1are given by ω=α12(e12 −e34) + α14e14 +α24e24, α12 = 0. All these two-forms are symplectomorphycally equivalent to either Ω1=e12 −e34 or Ω2=e12 −e34 +e24 through an automorphisms of d4,1, which are given by Φ =        z11 0 0 z14 0z22 0 0 z31 −z14z22 z11z22 z34 0 0 0 1        ,with z11z22 = 0. Besides, the automorphisms Φithat preserve each symplectic structure (d4,1,Ωi), i= 1,2, are determined by the conditions Φ1:z31 = 0, z22z11 = 1 and Φ2:z31 = 0, z22 = 1, z11 = 1. We study the two symplectic structures separately. 3.14.1 Para-Kähler structures on (d4,1,Ω1) There are three distinct situations that give rise to different geometries. 97 Four-dimensional Kähler and para-Kähler Lie groups Para-Kähler structures of non-zero constant para-holomorphic sectional curvature Assume that a43 = 0. Then necessarily a23 = 0 and a straightforward calculation shows that any para-Kähler structure is equivalent to (J11,⟨·,·⟩11) :    J11e1=−e1, J11e2=e2, J11e3=−κe4, J11e4=−1 κe3, ⟨·,·⟩11 = 2e1◦e2+κe3◦e3−1 κe4◦e4, κ = 0, so that d4,1splits as direct sums of Lagrangian subalgebras as d4,1=L11 ⊕L′ 11 = span{e2, e4−1 κe3}⊕span{e1, e4+1 κe3}. A straightforward calculation shows that the structures (J11,⟨·,·⟩11)have constant para-holomorphic sectional curvature H11 =κ, thus corresponding to those given by Theorem 3.1-(i.b). Locally symmetric Ricci-flat para-Kähler structures Set a43 = 0,a23 = 0. Then, any para-Kähler structure is equivalent to one of the following two families. (J12,⟨·,·⟩12) :    J12e1=−e1, J12e2=−κ e1+e2, J12e3=e3, J12e4=−e4, ⟨·,·⟩12 =κ e2◦e2+ 2(e1◦e2+e3◦e4), κ = 0,±1, (J13,⟨·,·⟩13) :    J13e1=e1+κ e2, J13e2=−e2, J13e3=e3, J13e4=−e4, ⟨·,·⟩13 =κ e1◦e1−2(e1◦e2−e3◦e4), κ = 0,±1. These structures induce the Lagrangian decompositions d4,1=L12 ⊕L′ 12 = span{e3, e2−κ 2e1}⊕span{e1, e4} =L13 ⊕L′ 13 = span{e3, e2+2 κe1}⊕span{e2, e4}(κ= 0) =L13 ⊕L′ 13 = span{e1, e3}⊕span{e2, e4}(κ= 0). The structures (J12,⟨·,·⟩12)are all flat, while the structures (J13,⟨·,·⟩13)are flat if and only if κ= 0. Otherwise, they are locally symmetric and Ricci-flat and so they correspond to those in Theorem 3.1-(i). As a consequence, their underlying pseudoRiemannian structures are modelled on (3.1) with Ψ(x1, x2) = ±(x2)2. Finally, (d4,1, J13,⟨·,·⟩13)do not admit any invariant opposite almost para-Kähler structures (see Remark 3.10). 98 3.15 Para-Kähler structures on d4,1 2 Locally symmetric para-Kähler structures with nilpotent Ricci operator Setting a43 = 0 and a23 = 0, it is non difficult to see that any para-Kähler structure is equivalent to (J14,⟨·,·⟩14) :    J14e1=κe2−e4, J14e2=e3, J14e3=e2, J14e4=−e1+κe3, ⟨·,·⟩14 =κ(e1◦e1+e4◦e4) + 2(e1◦e3+e2◦e4), κ ∈R, which correspond to the Lagrangian decompositions of d4,1 L14 ⊕L′ 14 = span{e2+e3, e4−e1−κe2}⊕span{e3−e2, e4+e1−κe2}. The structures (J14,⟨·,·⟩14)are locally symmetric and have two-step nilpotent Ricci operators. Furthermore, the metrics ⟨·,·⟩14 are locally conformally flat if and only if κ= 0. In any other case, their anti-self-dual Weyl curvature operators are two-step nilpotent, and so this structures correspond to those in Theorem 3.1-(ii.b). 3.14.2 Para-Kähler structures on (d4,1,Ω2) Direct calculations show that all the para-Kähler structures are flat in this case and they are equivalent to the structures (J21,⟨·,·⟩21) :    J21e1=−e1, J21e2=−κe1+e2, J21e3=e3, J21e4=−e4, ⟨·,·⟩21 =κe2◦e2+ 2(e1◦e2−e2◦e4+e3◦e4), κ ∈R, which correspond to the Lagrangian decompositions d4,1=L21 ⊕L′ 21 = span{e3, e2−κ 2e1}⊕span{e1, e4}. 3.15 Para-Kähler structures on d4,1 2 Let d4,1 2be the Lie algebra generated by {e1, e2, e3, e4}so that [e1, e2] = e3,[e1, e4] = −1 2e1,[e2, e4] = −1 2e2,[e3, e4] = −e3, whose automorphisms correspond to Φ =        z11 z12 0z14 z21 z22 0z24 z31 z32 z33 z34 0 0 0 1        ,with z31 = 2(z11z24 −z14z21), z32 = 2(z12z24 −z14z22), z33 =z11z22 −z12z21 = 0. 99 Four-dimensional Kähler and para-Kähler Lie groups According to [6], this Lie algebra corresponds to the semi-direct extension Re4⋉φ H3given by φ= ad(e4): φ(e1) = 1 2e1, φ(e2) = 1 2e2, φ(e3) = e3. All the symplectic forms on d4,1 2are given by ω=α12(e12 −e34) + α14e14 +α24e24, α12 = 0, so they are symplectomorphically equivalent to Ω = e12 −e34 (see [115]). To give a description of the para-Kähler structures on this Lie algebra, we make use of the Lie algebra automorphisms that preserve Ω, i.e., those which satisfy z11z22 −z12z21 = 1, z14z21 −z11z24 = 0, z14z22 −z12z24 = 0. Now, the description of para-Kähler structures depends on whether or not a43 = 0. Para-Kähler structures of non-zero constant para-holomorphic sectional curvature If a43 = 0, then any para-Kähler structures is equivalent to (J1,⟨·,·⟩1) :    J1e1=−e1, J1e2=e2, J1e3=−κe4, J1e4=−1 κe3, ⟨·,·⟩1= 2e1◦e2+κe3◦e3−1 κe4◦e4, κ = 0, which corresponds to the Lagrangian decompositions d4,1 2=L1⊕L′ 1= span{e2, e4−1 κe3}⊕span{e1, e4+1 κe3}. The para-holomorphic sectional curvatures of these structures are constant H1=κ, so they correspond to those given in Theorem 3.1-(i.b) Flat para-Kähler structures If a43 = 0, then the corresponding para-Kähler structures are flat and equivalent to (J2,⟨·,·⟩2) :    J2e1=e1, J2e2=−e2, J2e3=e3, J2e4=−e4, ⟨·,·⟩2= 2(−e1◦e2+e3◦e4), so that d4,1 2=L2⊕L′ 2= span{e3, e1}⊕span{e4, e2}. 100 3.17 Para-Kähler structures on d4,2 Flat para-Kähler structures Assuming that a21 =a23 =a41 = 0 and a22 = 0 as in Case 1, if a31 = 0 then the corresponding para-Kähler structures are equivalent to the flat para-Kähler structure given by (J31,⟨·,·⟩31) :    J31e1=−e1, J31e2=e2, J31e3=−e3, J31e4=e4, ⟨·,·⟩31 = 2(e1◦e4+e2◦e3), which induces the Lagrangian decomposition d4,2=L31 ⊕L′ 31 = span{e2, e4}⊕span{e1, e3}. The case where a31 = 0 will be considered bellow, since it gives rise to a different geometric situation. Ricci-flat para-Kähler structures Assuming that a21 =a23 =a41 =a22 = 0 as in Case 2, the corresponding paraKähler structures are equivalent to the structures (J32,⟨·,·⟩32)given by    J32e1=−e3, J32e2=−κe3+e4, J32e3=−e1, J32e4=−κe1+e2, ⟨·,·⟩32 =κ(e2◦e2+e4◦e4) + 2(e1◦e2+e3◦e4), κ ∈R, which correspond to the Lagrangian decomposition of d4,2as L32 ⊕L′ 32 = span{e3−e1, e4+e2−κe1}⊕span{e3+e1, e4−e2+κe1}. These structures are flat if κ= 0. Otherwise, they are Ricci-flat and special recurrent with recurrence one-forms ξ32 = 4e4and curvature operators R32(e2∧e4) = −3κe1∧e3=R32(e2, e4, e2, e4)e1∧e3. Consequently, the underlying structures are locally modelled on (3.1) and thus their curvature tensor is semi-symmetric and these structures are covered by Theorem 3.6. Finally, these structures do not admit opposite almost para-Kähler structures (see Remark 3.10). 107 Four-dimensional Kähler and para-Kähler Lie groups Para-Kähler structures with diagonalizable Ricci operator which are not semisymmetric Assuming that a21 =a23 = 0 and a41 = 0 as in Case 3, the para-Kähler structures are equivalent to (J33,⟨·,·⟩33) :    J33e1=−e1+κe4, J33e2=e2, J33e3=−e3, J33e4=e4, ⟨·,·⟩33 =κe1◦e1+ 2(e1◦e4+e2◦e3), κ = 0, so that d4,2=L33 ⊕L′ 33 = span{e2, e4}⊕span{e3, e4−2 κe1}. Their Ricci operators are diagonalizable with eigenvalues {4κ, 4κ, 0,0}and their self-dual and anti-self-dual Weyl curvature operators have the same eigenvalues. Moreover, W−has a double root of its minimal polynomial, and the curvature tensors are not semi-symmetric. These structures correspond to those in Theorem 3.7-(ii.a). The assumption that a21 = 0 and a23 = 0 as in Case 4 leads to para-Kähler structures equivalent to (J34,⟨·,·⟩34) :    J34e1=κ 2e4, J34e2=−1 κe3, J34e3=−κe2, J34e4=2 κe1, ⟨·,·⟩34 =κ 2e1◦e1+1 κe2◦e2−κe3◦e3−2 κe4◦e4, κ = 0, which correspond to the Lagrangian decompositions d4,2=L34 ⊕L′ 34 = span{e4+2 κe1, e3−κe2}⊕span{e4−2 κe1, e3+κe2}. Their Ricci operators are diagonalizable with eigenvalues {3κ, 3κ, 0,0}and their self-dual and the anti-self-dual Weyl curvature operators are diagonalizable with opposite eigenvalues. Moreover, the curvature tensor is not semi-symmetric and these structures correspond to those in Theorem 3.7-(ii.b). Remark 3.12.Let Q3ibe the almost product structures associated to the Ricci operators Ric3i, for i= 2,3, so that Q3i=−Id on ker Ric3iand Q3i= Id on the orthogonal distribution corresponding to the eigenspace of the non-zero Ricci curvature. These determine opposite almost para-complex structures defined by J′ 3i=J3iQ3i, i = 2,3. A straightforward calculation now shows that J′ 3iare opposite almost para-Kähler structures commuting with J3ithat have associated symplectic forms Ω′ 3i(x, y) = ⟨J′ 3ix, y⟩3i= Ω2, i = 3,4. 108 3.18 Kähler Lie algebras Para-Kähler structures which are not semi-symmetric with non-diagonalizable Ricci operator Assuming that a21 =a23 =a41 = 0 and a22 = 0 as in Case 1, if a31 = 0, the paraKähler structures on this Lie algebra are equivalent to the structures (J35,⟨·,·⟩35) given by    J35e1=−e1−2e3, J35e2= 2e4−e2−κe3, J35e3=e3, J35e4=e4, ⟨·,·⟩35 =κe2◦e2+ 2(e1◦e4+ 2e1◦e2−e2◦e3), κ ∈R, which induce the Lagrangian decompositions d4,2=L35 ⊕L′ 35 = span{e3, e4}⊕span{e4−e2+κ 2e1, e3+e1}. Their Ricci operators are two-step nilpotent and their curvature tensors are not semisymmetric. Moreover, their anti-self-dual Weyl curvature operator is three-step nilpotent and these structures correspond to those in Theorem 3.7-(i). Finally, if a21 = 0 as in Case 5, then the para-Kähler structures are equivalent to the structures (J36,⟨·,·⟩36)given by    J36e1=−e1+κ(e2+e4), J36e2=e2, J36e3=κ(e2+e4)−e3, J36e4=e4, ⟨·,·⟩36 =κ(e1◦e1+e3◦e3+ 2e1◦e3) + 2(e1◦e4+e2◦e3), κ = 0, so that d4,2=L36 ⊕L′ 36 = span{e2, e4}⊕span{e2−2 κe1+e4, e3−e1}. The Ricci operators of these structures have a single eigenvalue 3 2κ= 0 which is a double root of their minimal polynomials, and their anti-self-dual Weyl curvature operators are three-step nilpotent. Besides, their curvature tensors are not semisymmetric and these structures correspond to those in Theorem 3.7-(i). Since their anti-self-dual Weyl curvature operators are three-step nilpotent the structures above do not admit any opposite almost para-Kähler structure commuting with the Ricci operator [45]. 3.18 Kähler Lie algebras Kähler Lie algebras were classified by Ovando in [116] and the geometry of the corresponding structures, similar though it may be to that of para-Kähler Lie algebras, is more rigid, allowing less possibilities. The symmetric case is essentially the same, but there are no left-invariant locally symmetric Ricci-flat Kählerian structures in contrast with Theorem 3.1-(1.a). 109 Four-dimensional Kähler and para-Kähler Lie groups Theorem 3.13. Let (G, ⟨·,·⟩, J)be a non-flat locally symmetric four-dimensional indefinite Kähler Lie group. Then, it corresponds to one of the following situations. (1) Its Ricci operator is diagonalizable and one of the following holds: (1.a) Its holomorphic sectional curvature is a non-zero constant. (1.b) Its metric is Einstein with non-zero scalar curvature. (1.c) The underlying manifold is locally a product of two surfaces of constant Gaussian curvature. (2) Its Ricci operator is non-diagonalizable and one of the following holds: (2.a) Its Ricci operator has complex eigenvalues. (2.b) Its Ricci operator is two-step nilpotent. Following Ovando’s classification, the structures in Assertion (1.a) correspond to the Kähler structures on the Lie groups determined by d4,1 2and d′ 4,δ, where δ > 0. The Kähler structures in Assertion (1.b) correspond to the metrics ⟨·,·⟩ =a(e1◦e1−e2◦e2+e3◦e3−e4◦e4) on r′ 2with symplectic form Ω = a(e13 −e24). The Kähler structures which admit an opposite Kähler structure as in Assertion (1.c) correspond to the Kähler structures on rr3,0,r′ 4,0,δ for δ > 0, and the structures on r2r2given by the metrics ⟨·,·⟩ =a(e1◦e1+e2◦e2) + b(e3◦e3+e4◦e4) with the symplectic form Ω = ae12 +be34, for ab < 0. The Kähler structures in Assertion (2.a) correspond to the metrics ⟨·,·⟩ =a(e1◦e1−e2◦e2+e3◦e3−e4◦e4)+2b(e1◦e2+e3◦e4) on r′ 2with the symplectic form Ω = a(e13 −e24) + b(e14 +e23), for b= 0. Assertion (2.b) corresponds to the Kähler structures on d4,1. Remark 3.14.The metrics corresponding to Assertions (1.b) and (2.a) are linked by anti-Kähler structures, so that they have the same Levi-Civita connection as in the para-Kähler case. Moreover, amongst the structures above there are locally conformally flat Kähler Lie groups as stated in the following result. 110 3.18 Kähler Lie algebras Corollary 3.15. Let (M, g, J)be a locally conformally flat four-dimensional indefinite Kähler manifold. Then, it is flat or it is locally isometric to the Kähler Lie group determined by one of the following: (1) The Lie algebra r2r2with the metrics ⟨·,·⟩ =a(e1◦e1+e2◦e2−e3◦e3−e4◦e4) and the symplectic structure Ω = a(e12 −e34). (2) The Lie algebra r′ 2with the metrics ⟨·,·⟩ = 2b(e1◦e2+e3◦e4)and the symplectic structure Ω = b(e14 +e23). (3) The Lie algebra d4,1with the metrics ⟨·,·⟩ = 2a(e2◦e4−e1◦e3)and the symplectic structure Ω = a(e12 −e34). Finally, the non-symmetric case is significantly simpler than its para-Kähler counterpart since the existence of Kähler and opposite almost Kähler structures is much rigid than the corresponding para-Kähler analogue (see [44,72]). Theorem 3.16. Let (G, ⟨·,·⟩,Ω) be a non-symmetric four-dimensional indefinite Kähler Lie group. Then, one of the following holds. (1) (G, ⟨·,·⟩)is semi-symmetric if and only if its Ricci operator vanishes, in which case its curvature tensor is special recurrent and the metric is simply harmonic. (2) (G, ⟨·,·⟩)is not semi-symmetric if and only if it corresponds to the 3-symmetric space determined by the Kähler metrics ⟨·,·⟩ =a1 2e1◦e1+ 2e4◦e4+b(e2◦e2+e3◦e3) on d4,2with the symplectic form Ω = ae14 +be23, for ab < 0. Assertion (1) corresponds to the metrics ⟨·,·⟩ =−c(e1◦e1+e2◦e2)−2a(e1◦e4+e2◦e3)+2b(e1◦e3−e2◦e4) on r′ 2with the symplectic form Ω = ce12 +a(e13 −e24) + b(e14 +e23), where c(a2+b2)= 0, the metrics ⟨·,·⟩ =−c(e1◦e1+e4◦e4)+2b(e1◦e2−e3◦e4)−2a(e1◦e3+e2◦e4) on r4,−1,−1with symplectic the form Ω = a(e12 +e34) + b(e13 −e24) + ce14, where c(a2+b2)= 0, and the metrics ⟨·,·⟩ =b(e2◦e2+e4◦e4)+2a(e1◦e2+e3◦e4) on d4,2with the symplectic form Ω = be24 +a(e14 +e23), for a= 0. 111 Part II Solitons associated to geometric flows In this part, we will devote ourselves to the study of solitons associated to two particular geometric flows: the Ricci flow and the Bach flow. In Chapter 4 we will give a complete description of four-dimensional Lorentzian left-invariant Ricci solitons and in Chapter 5 we will introduce a general technique to approach the classification of algebraic solitons and give a complete classification of four-dimensional Riemannian both Ricci and Bach solitons. But before we start, we will briefly introduce the two geometric flows that are the subject of our study and their corresponding solitons. The Ricci flow: Ricci solitons The Ricci flow was introduced by Hamilton in [83] with the intention to solve the Poincaré conjecture, which declares that any three-dimensional closed and simply connected manifold is homeomorphic to S3. The Ricci flow is given by the evolution equation ∂ ∂t gt=−2ρgt,(II.1) where gtis a one-parameter family of pseudo-Riemannian metrics on a manifold M. For any given differentiable metric g0on a closed manifold M, there exists a unique solution gt, with t∈[0, ε)for some ε > 0, to the Ricci flow equation such that gt|t=0 =g0. The first examples of solutions to the Ricci flow are given by Einstein metrics, which provide solutions of the form gt= (1 −2µt)g0,where            t∈−∞,1 2µif µ > 0, t∈1 2µ,∞if µ < 0, t∈(−∞,∞)if µ= 0, for an initial Einstein metric g0such that ρg0=µg0. In any of the three cases given by the different values of µ,g0remains invariant modulo homotheties. If we allow the initial metric to change not only by homotheties but also by diffeomorphisms, a solution gtto the Ricci flow is said to be self-similar if there exists a positive function σ(t)and a one-parameter group of diffeomorphisms ψt:M→Msuch that gt=σ(t)ψ∗ tg0.(II.2) Remark 3.17.Assume that Equation (II.2) determines a solution to the Ricci flow and differentiate it to obtain ∂ ∂t gt=−2ρgt=dσ dt (t)ψ∗ tg0+σ(t)ψ∗ t(LXg0),(II.3) 115 where Ldenotes the Lie derivative and Xis the time-dependent vector field given by X(ψt(p)) = d dt(ψt(p)) for any p∈M. Now, since ρgt=ψ∗ tρg0, we can actually drop the pull-back in Equation (II.3) and so −2ρg0=dσ dt (t)g0+L˜ Xg0,(II.4) where ˜ X(t) = σ(t)X(t). If we now set µ=−1 2 dσ dt t=0 and X0=˜ X(0), Equation (II.4) becomes −2ρg0=−2µg0+ 2LX0g0 at t= 0. This proves that for any self-similar solution to the Ricci flow, there exists a vector field Xon Msuch that LXg+ρ=µg. Conversely, let Xbe a complete vector field on a pseudo-Riemannian manifold (M, g)and denote by ψt:M→Mthe family of diffeomorphisms generated by Xaccording to ∂ ∂t ψt(p) = 1 1−2µtX(ψt(p)) and ψ(0) = IdM, which is defined for all t∈(−∞,1 2µ)if µ > 0and for all t∈(1 2µ,∞)if µ < 0. If we now consider the one-parameter family of metrics gt= (1 −2µt)ψ∗ tg, then ∂ ∂t gt=−2µψ∗ tg+ (1 −2µ)ψ∗ tL1 1−2µtXg =ψ∗ t−2µg +LX(ψt(p))g. Now, if LXg+ρ=µg, then ∂ ∂t gt=ψ∗ t(−2ρ) = −2ψ∗ tρ=−2ρ(ψ∗ tg) = −2ρ(gt), and so gtis a solution to the Ricci flow. The remark above shows that there exists a correspondence between self-similar solutions to the Ricci flow and what are known as Ricci solitons. 116 5.5 Algebraic solitons on H3⋊ R P424 =−cT14 −dT24 −FT34, P431 =−dT13 +cT23, P432 =−cT13 −aT23, P433 = (a+d)(T44 −µ) + HT13 +FT23, P434 =−(a+d)T34. We start by considering P214 =−γT34, P211 +P323 −P424 =−2(γT13 −FT34), P212 −P313 +P414 =−2(γT23 +HT34), which imply that T13 =T23 =T34 = 0,(5.29) and split our analysis differentiating the cases a+d= 0 and a+d= 0. Case 1: a+d= 0. Using Equation (5.29), since P313 =−(a+d)T14, P323 =−(a+d)T24, P433 = (a+d)(T44 −µ), P213 =γ(T11 +T22 −T33 −µ)−FT14 +HT24, and a+d= 0 we obtain T14 =T24 = 0, T44 =µ, T33 =T11 +T22 −µ. (5.30) Next we show that the situation is different depending on whether cvanishes or not. Case 1.1: c= 0. In this case Equations (5.29) and (5.30) imply that P411 = 2cT12,P421 =−c(T11 −T22)+(a−d)T12, so the condition c= 0 leads to T12 = 0, T22 =T11.(5.31) 219 Algebraic solitons on four-dimensional Riemannian Lie groups At this point, Equations (5.29), (5.30) and (5.31) make b Tdiagonal, b T= diag[T11, T11,2T11 −µ, µ] and, moreover, the vanishing conditions of the divergence of Tgiven in Equation (5.28) reduce to (a+d)(T11 −µ) = 0. Hence, we conclude that there are no non-trivial algebraic T-soliton in this case. Case 1.2: c= 0. Using Equation (5.30) together with Equation (5.29) and c= 0, it is straightforward to see that T=         T11 T12 0 0 ∗T22 0 0 ∗ ∗ T11 +T22 −µ0 ∗ ∗ ∗ µ         . The system {Pijk = 0}now corresponds to P412 =−P421 =−(a−d)T12 = 0, P413 =FT12 −H(T22 −µ) = 0, P423 =−F(T11 −µ) + HT12 = 0, while the vanishing of the divergence of Tgiven by Equation (5.28) reduces to (2a+d)T11 + (a+ 2d)T22 −3(a+d)µ= 0. If a−d= 0, then T12 = 0. Besides, if FH = 0, then T11 =T22 =µand the tensor field Tis trivial. Thus, according to Remark 5.28, we can take F= 0 (working with an isomorphically isometric metric if necessary) and Assertion (i.a) immediately follows. If a=d, they are both different from zero, since a+d= 0. Therefore, the last equation above implies that T22 = 2µ−T11 while the other equations give FT12 +H(T11 −µ) = 0, HT12 −F(T11 −µ) = 0. Note that if either For His non-zero, then the corresponding algebraic T-soliton is trivial. Therefore, F=H= 0 and Assertion (i.b) follows. 220 5.5 Algebraic solitons on H3⋊ R Case 2: a+d= 0. In this situation, we distinguish two cases depending on whether a2and c2are equal or not. Case 2.1: a2=c2. A direct calculation involving Equation (5.29) and the condition d=−ashows that P211 =−cT14 +aT24, P212 =aT14 −cT24, 2aP411 +c(P412 −P421) = 2(a2−c2)(T44 −µ), P213 =γ(T11 +T22 −T33 −µ)−FT14 +HT24, so a2=c2implies that T14 =T24 = 0, T44 =µ, T33 =T11 +T22 −µ. (5.32) This last equation together with (5.29) lead to P411 = 2cT12,P421 = 2aT12 −c(T11 −T22), while the vanishing of the divergence of Tgiven in Equation (5.28) reduces to a(T11 −T22) = 0. Hence, since a2=c2, it follows that T12 = 0, T11 =T22.(5.33) Putting together Equations (5.29), (5.32) and (5.33) we obtain that b Tis diagonal, b T= diag[T11, T11,2T11 −µ, µ], and the system of polynomial equations {Pijk = 0}reduces to P413 =H(µ−T11) = 0,P423 =F(µ−T11) = 0. Thus, there are non-trivial algebraic T-solitons when T11 =µand F=H= 0, so Assertion (ii.a) is obtained. 221 Algebraic solitons on four-dimensional Riemannian Lie groups Case 2.2: a2=c2. We set c=εa, with ε2= 1, and assume that the conditions in Equation (5.29) hold. In this case, Equation (5.28), which gives the conditions for Tto be divergence-free, transforms into a(T11 −T22) = 0, a(εT14 −T24) = 0. If a= 0, then c=d= 0 and the tensor Tis divergence-free, while the system {Pijk = 0}reduces to P423 =F(T22 −T33 +T44 −µ) + HT12 −γT14 = 0, P413 =H(T11 −T33 +T44 −µ) + FT12 +γT24 = 0, P213 =γ(T11 +T22 −T33 −µ)−FT14 +HT24 = 0. Clearing T14 and T24 in the first and second equations above, respectively, Assertion (ii.b) is immediately obtained. If a= 0, then we compute P411 =a(2εT12 +T44 −µ), which together with the conditions for Tto be a divergence-free tensor give T22 =T11, T24 =εT14, T44 =µ−2εT12. Thus, the system of polynomial equations {Pijk = 0}reduces to P213 =γ(2T11 −T33 −µ)−(F−εH)T14 = 0, P413 =H(T11 −T33)+(F−2εH)T12 +εγT14 = 0, P423 =F(T11 −T33)+(H−2εF )T12 −γT14 = 0, and from the first equation above we obtain an expression for T33. Note that the corresponding left-invariant metrics are given by [e1, e2] = γe3,[e1, e4] = ae1−εae2+He3,[e2, e4] = εae1−ae2+F e3, and the isometry (e1, e2, e3, e4)7→ (e1,−e2,−e3, e4) interchanges (ε, a, F, H)and (−ε, a, F, −H). Thus we can take ε= 1 working, if necessary, with an isomorphically isometric metric and Assertion (ii.c) is obtained. 222 5.5 Algebraic solitons on H3⋊ R 5.5.2 Algebraic Ricci solitons on H3⋊ R A straightforward calculation shows that the Ricci tensor is determined by 2ρ11 =−4a(a+d)−H2−γ2,2ρ12 =−2c(a−d)−FH, 2ρ13 =−H(2a+ 3d) + Fc, 2ρ14 =Fγ, 2ρ22 =−4d(a+d)−F2−γ2,2ρ23 =−F(3a+ 2d)−Hc, 2ρ33 =−4(a+d)2+F2+H2+γ2,2ρ24 =−Hγ, 2ρ44 =−4(a2+d2+ad)−F2−H2. We use Theorem 5.29 to analyse all the possibilities for non-trivial algebraic Ricci solitons. In the locally symmetric case, Remark 5.27 guarantees that a=d=±1 2γ, F=H= 0, and a direct calculation shows that Ric = −3 2γ2 1Id, so the corresponding algebraic Ricci solitons are trivial. In the non-symmetric case, we study each of the five cases in Theorem 5.29 separately. Note that we can take γ= 1 and work in the same homothetic class. Case (i.a). We take c= 0,F= 0 and d=±a. Algebraic Ricci solitons must have diagonal Ricci operator, so the expression ρ24 =−1 2Himplies that H= 0. With this last condition, the Ricci operator is diagonal. Now, algebraic Ricci solitons are characterized by the vanishing of the polynomials Q1=ρ33 −ρ11 −ρ22 +µ, Q2=ρ44 −µ, Q3= (2a+d)ρ11 + (a+ 2d)ρ22 −3(a+d)µ, and a direct calculation shows that Q1=µ+3 2, Q2=−µ−2(a2+d2+ad), Q3=−1 2(a+d)6µ+ 8(a2+d2+ad)+3. Therefore the system {Qi= 0}is equivalent to 4(a2+d2+ad)−3 = 0, µ =−3 2, 223 Algebraic solitons on four-dimensional Riemannian Lie groups which determine algebraic Ricci solitons with associated left-invariant metrics given by [e1, e2] = e3,[e1, e4] = ae1,[e2, e4] = de2,[e3, e4]=(a+d)e3.(5.34) Note that the isometry e47→ −e4transforms (a, d)into (−a, −d), while the isometry (e1, e2, e3, e4)7→ (e2,−e1, e3, e4) transforms (a, d)into (d, a). As a consequence of this together with d=±a, we can assume that |a|< d. Besides, the condition 4(a2+d2+ad)−3 = 0 implies that a∈(−√3 2,1 2). This corresponds to Theorem 5.4-(iv) for a=−√3 2. Case (i.b). Since a=d= 0 and c=F=H= 0, a direct calculation shows that the Ricci tensor satisfies ρ33 =−8a2+1 2and ρ44 =−6a2. Since an algebraic Ricci soliton must satisfy ρ33 =ρ44 =µ, we get a=±1 2, which implies that the space is locally symmetric (see Remark 5.27). Case (ii.a). In this case, d=−a,a2=c2and F=H= 0. A direct calculation shows that ρ11 =ρ22 =−1 2and, moreover, the Ricci operator is diagonal if and only if ρ12 = 0, so the algebraic Ricci solitons are determined by the vanishing of the polynomials Q1=ρ12,Q2=ρ33 −2ρ11 +µ, Q3=ρ44 −µ. Computing these expressions we obtain Q1=−2ac, Q2=µ+3 2,Q3=−µ−2a2, which imply that c= 0 and a=±√3 2. With these conditions we obtain an algebraic Ricci soliton with associated left-invariant metric given by Equation (5.34) with a=−d=±√3 2. As we did in the previous case, we can set a=−√3 2, and this metric corresponds to Theorem 5.4-(iv) for this value of a. 224 5.5 Algebraic solitons on H3⋊ R Case (ii.b). This case has already been covered by the analysis on R3⋊ R (see Remark 5.30). Case (ii.c). In this last case d=−aand c=a= 0. Since ρ34 = 0 and ρ13 =−ρ23 =1 2a(F+H), then we must take H=−Fin order for the Ricci tensor to have the matrix form given in Theorem 5.29-(ii.c). Now, an algebraic Ricci soliton is characterized by the vanishing of the polynomials Q1=ρ11 −ρ22,Q2=ρ33 −2ρ11 + 2Fρ14 +µ, Q3=ρ44 + 2ρ12 −µ, Q4=ρ24 −ρ14, Q5=−Fρ11 −3Fρ12 + (2F2−1)ρ14 +Fµ, and a direct calculation leads to Q1=Q4= 0 and Q2=µ+ 3F2+3 2,Q3=−µ−6a2,Q5=F(µ+ 6a2).(5.35) Consequently, 4a2−2F2−1 = 0 and it follows from Remark 5.30 that this case has already been covered by the analysis on R3⋊ R. Remark 5.31.For the sake of completeness, we will include here the solution of the two cases covered by the analysis on R3⋊ R. Case (ii.b) in Theorem 5.29 is given by a=c=d= 0. In this case, it is easy to check that ρ13 =ρ23 =ρ34 = 0, so the matrix form in Theorem 5.29-(ii.b) is satisfied and an algebraic Ricci soliton is determined by the vanishing of the polynomials Q1=ρ14 −F(ρ22 −ρ33 +ρ44 −µ)−Hρ12, Q2=ρ24 +H(ρ11 −ρ33 +ρ44 −µ) + Fρ12, Q3= (H2−1)ρ11 + (F2−1)ρ22 −(F2+H2−1)ρ33 + (F2+H2)ρ44 + 2FHρ12 −(F2+H2−1)µ. A direct calculation gives Q1=1 2F(3F2+ 3H2+ 2µ+ 3), Q2=−1 2H(3F2+ 3H2+ 2µ+ 3), Q3=−1 2(F2+H2−1)(3F2+ 3H2+ 2µ+ 3), 225 Algebraic solitons on four-dimensional Riemannian Lie groups so the system {Qi= 0}is equivalent to µ=−3 2(F2+H2+ 1). The associated left-invariant metrics are given by [e1, e2] = e3,[e1, e4] = He1,[e2, e4] = Fe3,(5.36) and considering the orthogonal basis ˜e1=−2 F2+H2+1 (Fe1−He2+e4), ˜e2=√2 √(F2+H2+1)(F2+H2)He1+Fe2−√F2+H2e3, ˜e3=−√2 √(F2+H2+1)(F2+H2)He1+Fe2+√F2+H2e3, ˜e4=2 (F2+H2+1)√F2+H2Fe1−He2−(F2+H2)e4, the only non-zero brackets are [˜e2,˜e4] = ˜e2+ ˜e3and [˜e3,˜e4] = −˜e2−˜e3. Moreover, ⟨˜ei,˜ej⟩=4 F2+H2+1 ⟨ei, ej⟩, which shows that the above metric Lie groups are isomorphically homothetic to the metric in Theorem 5.4-(i). Finally, in Case (ii.c) of Theorem 5.29, using Equation (5.35) we obtain that 4a2−2F2−1 = 0 and µ=−6a2are the conditions that determine the algebraic Ricci solitons in this case. These have associated left-invariant metrics given by [e1, e2] = e3,[e1, e4] = ae1−ae2−Fe3,[e2, e4] = ae1−ae2+Fe3.(5.37) Considering the orthogonal basis ˜e1=1 4a2(Fe1+Fe2+ 2ae3+e4), ˜e2=−1 4a2√2((2a−F)e1−(2a+F)e2+ 2ae3−e4), ˜e3=−1 4a2√2((2a+F)e1−(2a−F)e2−2ae3+e4), ˜e4=1 4a2(e1+e2−2Fe4), the non-zero brackets correspond to [˜e1,˜e4] = 1 √2(˜e2+ ˜e3),[˜e2,˜e4] = −1 √2˜e1+ ˜e2,[˜e3,˜e4] = −1 √2˜e1−˜e3, and ⟨˜ei,˜ej⟩=1 2a2⟨ei, ej⟩. Hence, the metric Lie groups above are isomorphically homothetic to the metric given in Assertion (ii) of Theorem 5.4. 226 5.5 Algebraic solitons on H3⋊ R 5.5.3 Algebraic Bach solitons on H3⋊ R A long but straightforward calculation shows that the components Bij of the Bach tensor of H3⋊ R are determined by 24B11 =−16a3d+ 48ad3+ 84a2c2+ 16a2d2−108c2d2+ 24ac2d + (F2−20H2−20γ2)a2−21(F2−H2)c2−3(4F2+ 19H2+ 4γ2)d2 + 78FHac −4(22H2+ 7γ2)ad + 78FHcd −4(F2+H2+γ2)(F2−3H2−3γ2), 12B12 =−18a3c+ 24ac3−24c3d+ 18cd3−58a2cd + 58acd2 −3FH 4a2−7c2+ 4d2+ (31F2−8H2−2γ2)ac −53FHad + (8F2−31H2+ 2γ2)cd + 8FH(F2+H2+γ2), 12B13 =−3Fc3−9Hd3+ 33Fa2c+ 3Hac2−28Ha2d−48Had2+ 24Hc2d −9Fcd2+ 53Facd + 8(F2+H2+γ2)(2Ha −Fc + 3Hd), 12B14 =γ3Fa2−c2−3Hac + 14Fad + 15Hcd −8F(F2+H2+γ2), 24B22 = 48a3d−16ad3−108a2c2+ 16a2d2+ 84c2d2+ 24ac2d −3(19F2+ 4H2+ 4γ2)a2+ 21(F2−H2)c2−(20F2−H2+ 20γ2)d2 −78FHac −4(22F2+ 7γ2)ad −78FHcd + 4(F2+H2+γ2)(3F2−H2+ 3γ2), 12B23 =−9Fa3+ 3Hc3+ 9Ha2c+ 24Fac2−48Fa2d−28Fad2+ 3Fc2d −33Hcd2−53Hacd + 8(F2+H2+γ2)(3Fa +Hc + 2Fd), 12B24 =γ3Hc2−d2+ 15Fac −14Had −3Fcd + 8H(F2+H2+γ2), 24B33 =−16a3d−16ad3−12a2c2−48a2d2−12c2d2+ 24ac2d + (43F2+ 28H2+ 28γ2)a2−9(F2+H2)c2+ (28F2+ 43H2+ 28γ2)d2 −54FHac+(104F2+104H2+44γ2)ad+54FHcd−20(F2+H2+γ2)2. Recall that the Bach tensor is trace-free, so B44 =−B11 −B22 −B33. We study the existence of non-trivial algebraic Bach solitons using Theorem 5.29. In the locally symmetric case, the underlying manifold is homothetic to the complex hyperbolic plane (see Remark 5.27), and so it is self-dual, and thus Bach-flat. In the non-symmetric case, we proceed as in the previous section by checking the five cases given in Theorem 5.29 separately. Without loss of generality we can set γ= 1 remaining in the same homothetic class. Case (i.a). In this first case, c= 0,F= 0 and d=±a. Note that B12 =B14 =B23 =B34 = 0 227 Algebraic solitons on four-dimensional Riemannian Lie groups and recall that B44 =−B11 −B22 −B33. If we now impose the diagonal form and the conditions in Theorem 5.29-(i.a) to the Bach operator we have that an algebraic Bach soliton is determined by the vanishing of the polynomials Q1=B13,Q2=B24,Q3=−2(B11 +B22), Q4=H(B11 + 2B22 +B33), Q5= (5a+ 4d)B11 + (4a+ 5d)B22 + 3(a+d)B33, Q6=µ+B11 +B22 +B33. First of all we use Q2and Q3to show that Hmust be zero. Since Q2=−1 12H(3d2+ 14ad −8H2−8) must vanish, if H= 0 then dcannot be null and a=−1 14d(3d2−8H2−8). With this value for aa long but direct calculation shows that Q3=1 223636d2d2+ 2H2+ 2652H2(407d2+ 64H2+ 128) + (326d2−83)2+ 34839 which leads to a contradiction since Q3= 0 and none of the factors above vanishes. Consequently H= 0 and the system {Qi= 0}becomes Q3=−2 3(ad −1)(4(a2+d2+ad)−3), Q5=2 3(ad −1)(4(a2+d2+ad)−3)(a+d), Q6=µ+1 6(4ad −1)(a2+d2−ad −1), or equivalently, µ=−1 6(4ad −1)(a2+d2−ad −1),(ad −1)(4(a2+d2+ad)−3) = 0. Thus, we obtain algebraic Bach solitons with associated left-invariant metrics given by [e1, e2] = e3,[e1, e4] = ae1,[e2, e4] = de2,[e3, e4] = (a+d)e3. If 4(a2+d2+ad)−3 = 0, the space is an algebraic Ricci soliton (see Equation (5.34)). Otherwise, ad = 1, so these structure constants must be non-zero and d=1 a. In this case, µ=−(a2−1)2 2a2, the associated left-invariant metrics transform into [e1, e2] = e3,[e1, e4] = ae1,[e2, e4] = 1 ae2,[e3, e4] = a2+1 ae3, 228 Jensen proved in [88] a well-known result that states that, in the Riemannian homogeneous setting, Einstein metrics are symmetric, which constitutes a very rigid situation. The Weyl curvature tensor of any Einstein metric is divergence-free – in other words, it is harmonic –, a condition that is equivalent to the fact that the corresponding Schouten tensor is a Codazzi tensor. If the scalar curvature is constant, then the Ricci tensor is also a Codazzi tensor. Podestà and Spiro showed in [123] that the condition div W= 0 implies local symmetry in the locally homogeneous four-dimensional case. On four-dimensional oriented four-manifolds, the decomposition of the Weyl curvature tensor into its self-dual and anti-self-dual components W=W++W−, which is conformal invariant, makes the harmonicity condition div W= 0 become div W++ div W−= 0. A four-dimensional manifold is said to have half-harmonic Weyl curvature tensor if either div W+= 0 or div W−= 0. These conditions are natural generalizations of the Einstein condition and have received special attention during the last decade (see [46,48,107] and [134]). In Theorem 6.1 we give the complete description – up to homotheties – of fourdimensional locally homogeneous Riemannian manifolds with half-harmonic Weyl curvature tensor. The manifolds in this result are naturally equipped with a selfdual homogeneous structure which is not necessarily the canonical homogeneous structure of the underlying Lie group. Given the fact that homogeneous manifolds may admit different presentations as homogeneous spaces, they may as well admit more than one homogeneous structure. Therefore we decided to study which Lie groups admit at least two inequivalent homogeneous structures. In Theorem 7.1 we show that a three-dimensional non-symmetric homogeneous space admits at least two inequivalent homogeneous structures if and only if its isometry group is fourdimensional. Motivated by the results obtained in Theorem 6.1, we started to investigate fourdimensional self-dual homogeneous structures, obtaining some partial classification results. Nevertheless, the whole classification of four-dimensional Riemannian selfdual homogeneous structures remains an open problem. 235 Chapter 6 Homogeneous four-manifolds with half-harmonic Weyl curvature In this chapter we will work in the Riemannian homogeneous setting to study what the manifolds that admit half-harmonic Weyl curvature are like. The contents of this chapter are partially contained in the work [37]. 6.1 Summary of results A four-dimensional Riemannian manifold (M, g)has harmonic Weyl curvature if its Weyl tensor is divergence free, i.e., δW = div1W= 0 or, equivalently, if its Cotton tensor vanishes. Even though the Weyl curvature tensor is conformally invariant, the condition δW = 0 is not. In fact, for any conformal metric ˜g=e2σg, one has that f div1f W= div1W−ι∇σW. Einstein metrics have harmonic Weyl curvature, which is the reason why the condition δW = 0 has been investigated in order to extend the geometric properties of Einstein metrics to more general structures. On the other hand, any locally symmetric manifold trivially has harmonic Weyl curvature and it follows from [15, 54] that the converse is also true in the four-dimensional homogeneous setting (see also [123]). We have already mentioned that the space of two-forms on an oriented fourdimensional Riemannian manifold splits as the direct sum of the spaces of self-dual and anti-self-dual two-forms, Λ2 ±(TM), under the action of SO(4) and that such decomposition corresponds to the one given by the ±1eigenspaces of the Hodge-star operator ⋆: Λ2(TM)→Λ2(TM), so that the self-dual and anti-self-dual components of any two-form Ωare given by Ω±=1 2(Ω ±⋆Ω). Since the Weyl conformal curvature operator W ∈ End Λ2(TM) and the Hodge-star operator commute, the bundles Λ2 ±(TM)remain invariant under its action. In what follows, we will denote by W±the self-dual and anti-self-dual components of the Weyl curvature operator, which are its restrictions to the self-dual and anti-self-dual subspaces of the space of two-forms, respectively, and an oriented Riemannian four-manifold is said to be self-dual (resp. anti-self-dual) if W−= 0 237 Homogeneous four-manifolds with half-harmonic Weyl curvature (resp. W+= 0). Given the decomposition of the Weyl curvature operator into its self-dual and anti-self-dual components, its divergence decomposes accordingly and so (M, g)is said to have half-harmonic Weyl curvature if either δW+= 0 or δW−= 0. The conditions δW±= 0 have been extensively investigated during the past decade in order to describe different classes of Riemannian manifolds such as Ricci solitons [46,48,134] and quasi-Einstein metrics [107]. A first example of metrics that have half-harmonic Weyl curvature can be found in the Kähler setting. The geometry of four-dimensional Kähler manifolds is, to a great extent, codified by their self-dual and anti-self-dual Weyl curvatures. In fact, the anti-self-dual Weyl curvature codifies the Bochner tensor and so the condition δW−= 0 is equivalent to weak Bochner-flatness (δB = 0) [8,87]. It was shown in [89] that weakly Bochner-flat Kähler surfaces of constant scalar curvature are locally symmetric (thus having harmonic Weyl curvature). On the other hand, the self-dual component of the Weyl operator is W+=τ 12 diag[2,−1,−1] and τdiv1W++ι∇τW+= 0. Therefore, δW+= 0 if and only if the scalar curvature τis constant. Moreover, any four-dimensional Kähler surface with W+= 0 is conformal to a metric with δW+= 0. We will show that four-dimensional homogeneous manifolds with half-harmonic Weyl curvature tensor are either symmetric or locally homothetic to the only nonsymmetric anti-self-dual homogeneous manifold or to the 3-symmetric space, so nonsymmetric homogeneous four-manifolds with half-harmonic Weyl curvature have an underlying complex structure that is either Kähler or locally conformally Kähler. This description is summarised in the main result of this chapter – whose proof is detailed in Section 6.2 – as follows. Theorem 6.1. Let (M, g)be a four-dimensional locally homogeneous Riemannian manifold with half-harmonic Weyl curvature tensor. Then it is symmetric or locally homothetic to one of the following semi-direct extensions of the Heisenberg group: (i) The left-invariant metric on H3⋊ R determined by [e1, e2] = e3,[e1, e4] = −e1,[e2, e4] = −e2,[e3, e4] = −2e3, (ii) or the left-invariant metric on H3⋊ R determined by [e1, e2] = e3,[e1, e4] = 1 2e1,[e2, e4] = −e2,[e3, e4] = −1 2e3, where {e1, e2, e3, e4}is an orthonormal basis. 238 6.1 Summary of results Remark 6.2.The metrics in Theorem 6.1-(i) are anti-self-dual and correspond to those in [55]. The anti-self-dual Weyl curvature operator has eigenvalues {−2,1,1} and Ω−=e1∧e2−e3∧e4is an eigenvector associated to the distinguished eigenvalue −2. A straightforward calculation shows that the associated almost complex structure J−e1=e2,J−e3=−e4is integrable and dΩ−=−e4∧Ω−. Therefore, the structure is locally conformally Kähler with respect to the opposite orientation. Furthermore, it was shown in [39] that these metrics are conformally Ricci-flat and, therefore, quasiEinstein. Remark 6.3.The self-dual and anti-self-dual Weyl curvature operators of the metrics in Theorem 6.1-(ii) have opposite eigenvalues {±1 4,∓1 2,±1 4}. The eigenvectors corresponding to the distinguished eigenvalues of multiplicity one, Ω±=e1∧e3∓e2∧e4, define a symplectic pair so that the underlying manifold is Kähler and opposite almost Kähler. While Ω+determines a Kähler structure J+e1=e3,J+e2=−e4, the opposite almost complex structure J−e1=e3,J−e2=e4determined by Ω−is not integrable and the Ricci operator is J−-invariant. Now it follows from [7, Theorem 1] that it corresponds to the unique four-dimensional 3-symmetric space, which is the only homogeneous non-symmetric Kähler surface. Moreover, it is an algebraic Ricci soliton [42], thus being an expanding Ricci soliton. Remark 6.4.The Bach tensor, B= div2div4W+1 2W[ρ], of the metrics given by Theorem 6.1-(i) vanishes, since they are anti-self-dual. Therefore, these metrics are trivially steady Bach solitons (see [81]). On the contrary, the Bach tensor of the Kähler metrics in Theorem 6.1-(ii), B= div2div4W+1 2W[ρ] = div2div4W++1 2W+[ρ] = 1 2W+[ρ], is determined by the non-zero components B(e1, e1) = B(e3, e3) = −B(e2, e2) = −B(e4, e4) = −3 8. A straightforward calculation shows that D=ˆ B−3 8Id is a derivation of the Lie algebra in Theorem 6.1-(ii), where ˆ Bis the (1,1)-tensor field associated to the (0,2)-Bach tensor B. Let φtbe the one-parameter family of automorphisms of the Lie algebra determined by dφt|e= e−t 2D. Proceeding as in Lauret’s work [97], the vector field Xon H3⋊ R given by X=d dt |t=0 φtdefines a Bach soliton, i.e., LX⟨·,·⟩ +B=3 8⟨·,·⟩. Consequently, the 3-symmetric space is both an algebraic Ricci soliton and an algebraic Bach soliton. Moreover, since the metric does not 239 Homogeneous four-manifolds with half-harmonic Weyl curvature split as a product Rk×N4−k, neither the Ricci nor the Bach solitons are gradient (see [81,121]). The soliton above does not correspond to those studied in [81] since the underlying structure is not the product H3×R. Remark 6.5.The norm of the self-dual Weyl curvature operator of any Kähler surface satisfies ∥W+∥2=1 24τ2. If the scalar curvature is constant, then it follows from the Weitzenböck formula (see [18]) ∆∥W+∥2=−τ∥W+∥2+ 36 detΛ2 +W+−2∥∇W+∥2 that the self-dual Weyl curvature operator is parallel. Therefore, homogeneous fourmanifolds with half-harmonic Weyl curvature are such that ∇W+= 0. Remark 6.6.The Bach tensor of the anti-self-dual metric in Theorem 6.1-(i) vanishes and so it is critical for the functional Ftwith t=−1 3. The energy of this functional is strictly negative and this metric is the only one which is Bach-flat with non-zero energy on a semi-direct extension of the Heisenberg group. The 3-symmetric space in Theorem 6.1-(ii) is a Ricci soliton and thus it is critical for the functional Ftwith t=−1 2. This is the only metric which is critical with zero energy on a semi-direct extension of the Heisenberg group. 6.2 Half-harmonic Weyl curvature on homogeneous fourmanifolds Simply connected four-dimensional homogeneous Riemannian manifolds are either symmetric or isometric to a Lie group with a left-invariant metric (see [15]). Any non-symmetric homogeneous metric is thus realized either on the product Lie groups SU(2)×Rand f SL(2,R)×R, or on the semi-direct extensions E(1,1)⋊R,e E(2)⋊R, H3⋊ R and R3⋊ R, where E(1,1),e E(2),H3and R3denote the Poincaré group, the the universal covering of the Euclidean group, the Heisenberg group and the Abelian group, respectively. As we have already mentioned, any symmetric four-dimensional homogeneous Riemannian manifold has harmonic Weyl curvature, therefore this situation will not provide examples of strictly half-harmonic Weyl curvature. De Smedt and Salamon proved that if (G, g)is a simply connected Lie group with a left-invariant metric and {e1, e2, e3, e4}is an orthonormal basis of its Lie algebra, there are two cases in which there exist orientation-reversing isometric automorphisms (see [55, Lemma 2.2]). 1. The Lie algebra ghas non-zero centre zand the isometric automorphism that reverses the orientation of (G, g)is given by (e1, e2, e3, e4)7−→ (−e1, e2, e3, e4) 240 6.2 Half-harmonic Weyl curvature on homogeneous four-manifolds for an orthonormal basis such that e1∈z. This situation corresponds to the products f SL(2,R)×Rand SU(2) ×R. 2. Gcorresponds to the semi-direct extension R3⋊ R. In this situation, if we fix an orthonormal basis such that e2, e3, e4∈r3, then (e1, e2, e3, e4)7−→ (e1,−e2,−e3,−e4) determines an orientation-reversing isometric automorphism of G. As a consequence, the Weyl curvature of these three Lie groups cannot be strictly half-harmonic. Nevertheless, we will address these cases quickly in the following remarks for the sake of completeness. Remark 6.7.We will use a direct approach to show that the Weyl curvature of R3⋊ R cannot be strictly half-harmonic. Let us consider left-invariant metrics on the semidirect extensions of the Abelian Lie algebra as described in Section 1.4.2. In this case there exists an orthonormal basis {e1, e2, e3, e4}such that [e1, e4] = ae1+be2+ce3,[e2, e4] = −be1+fe2+he3, [e3, e4] = −ce1−he2+pe3. (6.1) Now, a straightforward calculation shows that the divergences div1W±(ei, ej, ek) are determined – up to the corresponding symmetries – by div1W±(ei, ej, ek) = 1 4P± ijk, where P± ijk are the polynomials on the structure constants in (6.1) given by P± 112 =∓c(a2−2p2+af +ap −fp)−bh(a−2f+p), P± 113 =±b(a2−2f2+af +ap −fp) + ch(a+f−2p), P± 114 =a2(f+p)−af2−ap2−2(b2+c2)a+ 2b2f+ 2c2p, P± 212 =∓h(f2−2p2+af −ap +fp)−bc(2a−f−p), P± 213 =±a2f−(a+p)f2+fp2−2b2a+ 2(b2+h2)f−2h2p, P± 214 =−b(2a2−f2−af +ap −fp)−ch(a+f−2p), P± 312 =∓a2p+f2p−(a+f)p2−2c2a−2h2f+ 2(c2+h2)p, P± 313 =±h(2f2−p2+af −ap −fp)−bc(2a−f−p), P± 314 =−c(2a2−p2+af −ap −fp) + bh(a−2f+p), 241 Homogeneous four-manifolds with half-harmonic Weyl curvature P± 412 =−b(a−f)2, P± 413 =−c(a−p)2, P± 414 =∓h(f−p)2. Consequently, the conditions δW+= 0 and δW−= 0 are equivalent. Remark 6.8.As in the previous remark, we will make a direct approach to show that the Weyl curvatures of f SL(2,R)×Rand SU(2)×Rcannot be strictly half-harmonic. Let us consider left-invariant metrics on the product Lie groups f SL(2,R)×R and SU(2) ×Ras described in Section 1.4.2. We recall that in this case there exists an orthonormal basis {e1, e2, e3, e4}of the Lie algebra sl(2,R)×Ror su(2) ×R such that [e1, e2] = λ3e3,[e1, e3] = −λ2e2,[e2, e3] = λ1e1, [e1, e4] = k3λ2e2−k2λ3e3,[e2, e4] = k1λ3e3−k3λ1e1, [e3, e4] = k2λ1e1−k1λ2e2, (6.2) where λ1λ2λ3= 0. The associated Lie group corresponds to SU(2) ×Rif λ1,λ2, λ3do not change sign, and to f SL(2,R)×Rotherwise. In this case, a long but straightforward calculation shows that the divergence div1W+(ei, ej, ek)is determined – up to the corresponding symmetries – by div1W+(ei, ej, ek) = 1 16P+ ijk, where P+ ijk are the polynomials on the structure constants in (6.2) given by P+ 112 =−22λ3 1−λ3 3−λ2 1λ3k3 2+ 2 λ3 3−2λ1λ2 2+λ1λ2λ3k2 1k2 −4λ3 1−λ2 1(λ2+λ3)−λ2 2(λ1+ 2λ3) + λ1λ2λ3k2k2 3 +λ3λ2 1−λ2 2−3λ1λ2+ 4λ1λ3−λ2λ3k1k3 −4λ3 1−2λ3 3−λ2 1(λ2+ 2λ3)−λ2 2(λ1−λ3) + λ2λ2 3k2, P+ 113 =−22λ3 1−λ3 2−λ2 1λ2k3 3+ 2 λ3 2−2λ1λ2 3+λ1λ2λ3k2 1k3 −4λ3 1−λ2 1(λ2+λ3)−λ2 3(λ1+ 2λ2) + λ1λ2λ3k2 2k3 −λ2λ2 1−λ2 3+ 4λ1λ2−3λ1λ3−λ2λ3k1k2 −4λ3 1−2λ3 2−λ2 1(2λ2+λ3) + λ2 2λ3−λ2 3(λ1−λ2)k3, P+ 114 =λ2 1(λ2−λ3) + λ2 2(4λ1+ 3λ3)−λ2 3(4λ1+ 3λ2)k1k2k3 + 2 λ3 2+λ3 3−λ2 2λ3−λ2λ2 3k2 1 −4λ3 1−2λ3 3−λ2 1(λ2+ 2λ3) + λ2λ2 3k2 2 −4λ3 1−2λ3 2−λ2 1(2λ2+λ3) + λ2 2λ3k2 3 242 6.2 Half-harmonic Weyl curvature on homogeneous four-manifolds −22λ3 1−λ3 2−λ3 3−λ2 1(λ2+λ3) + λ2 2λ3+λ2λ2 3, P+ 212 = 2 2λ3 2−λ3 3−λ2 2λ3k3 1−2λ3 3−2λ2 1λ2+λ1λ2λ3k1k2 2 +4λ3 2−λ2 1(λ2+ 2λ3)−λ2 2(λ1+λ3) + λ1λ2λ3k1k2 3 −λ3λ2 1−λ2 2+ 3λ1λ2+λ1λ3−4λ2λ3k2k3 +4λ3 2−2λ3 3−λ2 1(λ2−λ3)−λ2 2(λ1+ 2λ3) + λ1λ2 3k1, P+ 213 =λ2 1(4λ2+ 3λ3) + λ2 2(λ1−λ3)−λ2 3(3λ1+ 4λ2)k1k2k3 +4λ3 2−2λ3 3−λ2 2(λ1+ 2λ3) + λ1λ2 3k2 1 −2λ3 1+λ3 3−λ2 1λ3−λ1λ2 3k2 2 −2λ3 1−4λ3 2−λ2 1λ3+λ2 2(2λ1+λ3)k2 3 −2λ3 1−2λ3 2+λ3 3−λ2 1λ3+λ2 2(λ1+λ3)−λ1λ2 3, P+ 214 = 2 λ3 1−2λ3 2+λ1λ2 2k3 3 −4λ3 2−λ2 2(λ1+λ3)−λ2 3(2λ1+λ2) + λ1λ2λ3k2 1k3 + 2 λ3 1−2λ2λ2 3+λ1λ2λ3k2 2k3 +λ1λ2 2−λ2 3+ 4λ1λ2−λ1λ3−3λ2λ3k1k2 +2λ3 1−4λ3 2−λ2 1λ3+λ2 2(2λ1+λ3)−λ2 3(λ1−λ2)k3, P+ 312 =λ2 1(3λ2+ 4λ3)−λ2 2(3λ1+ 4λ3) + λ2 3(λ1−λ2)k1k2k3 +2λ3 2−4λ3 3−λ1λ2 2+λ2 3(λ1+ 2λ2)k2 1 +2λ3 1−4λ3 3−λ2 1λ2+λ2 3(2λ1+λ2)k2 2 + 2 λ3 1+λ3 2−λ2 1λ2−λ1λ2 2k2 3 + 2 λ3 1+λ3 2−2λ3 3−λ2 1λ2−λ1λ2 2+λ2 3(λ1+λ2), P+ 313 =−2λ3 2−2λ3 3+λ2λ2 3k3 1 +4λ3 3−λ2 1(λ3+ 2λ2)−λ2 3(λ1+λ2) + λ1λ2λ3k1k2 2 −2λ3 2−2λ2 1λ3+λ1λ2λ3k1k2 3 +λ2λ2 1−λ2 3+λ1λ2+ 3λ1λ3−4λ2λ3k2k3 −2λ3 2−4λ3 3−λ2 1(λ2−λ3)−λ1λ2 2+λ2 3(λ1+ 2λ2)k1, P+ 314 =−2λ3 1−2λ3 3+λ1λ2 3k3 2 +4λ3 3−λ2 2(λ3+ 2λ1)−λ2 3(λ1+λ2) + λ1λ2λ3k2 1k2 −2λ3 1−2λ2 2λ3+λ1λ2λ3k2k2 3 −λ1λ2 2−λ2 3+λ1λ2−4λ1λ3+ 3λ2λ3k1k3 −2λ3 1−4λ3 3−λ2λ2 1−λ2 2(λ1−λ3) + λ2 3(2λ1+λ2)k2, P+ 412 =−2λ3 1+λ3 2−λ2 1λ2−λ1λ2 2k3 3 −2λ3 2−λ2 2(λ1+λ3) + λ2 3(2λ1−λ2)−λ1λ2λ3k2 1k3 −2λ3 1−λ2 1(λ2+λ3)−λ2 3(λ1−2λ2)−λ1λ2λ3k2 2k3 +λ2 1(3λ2−λ3)−λ2 2(3λ1−λ3)−λ2 3(λ1−λ2)k1k2 243 Homogeneous four-manifolds with half-harmonic Weyl curvature −2λ3 1+λ3 2−λ2 1λ2−λ1λ2 2k3, P+ 413 = 2 λ3 1+λ3 3−λ2 1λ3−λ1λ2 3k3 2 +2λ3 3+λ2 2(2λ1−λ3)−λ2 3(λ1+λ2)−λ1λ2λ3k2 1k2 +2λ3 1−λ2 1(λ2+λ3)−λ2 2(λ1−2λ3)−λ1λ2λ3k2k2 3 −λ2 1(λ2−3λ3) + λ2 2(λ1−λ3) + λ2 3(3λ1−λ2)k1k3 + 2 λ3 1+λ3 3−λ2 1λ3−λ1λ2 3k2, P+ 414 =−2λ3 2+λ3 3−λ2 2λ3−λ2λ2 3k3 1 −2λ3 3+λ2 1(2λ2−λ3)−λ2 3(λ1+λ2)−λ1λ2λ3k1k2 2 −2λ3 2−λ2 1(λ2−2λ3)−λ2 2(λ1+λ3)−λ1λ2λ3k1k2 3 −λ2 1(λ2−λ3) + λ2 2(λ1−3λ3)−λ2 3(λ1−3λ2)k2k3 −2λ3 2+λ3 3−λ2 2λ3−λ2λ2 3k1. For its part, div1W−(ei, ej, ek)is given – up to the corresponding symmetries – by div1W−(ei, ej, ek) = 1 16P− ijk, where P− ijk are polynomials on the structure constants in (6.2) that can be expressed in terms of the polynomials P+ ijk as follows. P− 112 =−P+ 112 + 2k1k3λ3λ2 1−λ2 2−3λ1λ2+ 4λ1λ3−λ2λ3, P− 113 =−P+ 113 −2k1k2λ2λ2 1−λ2 3+ 4λ1λ2−3λ1λ3−λ2λ3, P− 114 =−P+ 114 + 2k1k2k3(λ2−λ3)λ2 1+ 4λ1λ2+ 4λ1λ3+ 3λ2λ3, P− 212 =−P+ 212 −2k2k3λ3λ2 1−λ2 2+ 3λ1λ2+λ1λ3−4λ2λ3, P− 213 =P+ 213 −2k1k2k3(λ1−λ3)λ2 2+ 4λ1λ2+ 3λ1λ3+ 4λ2λ3, P− 214 =P+ 214 −2k1k2λ1λ2 2−λ2 3+ 4λ1λ2−λ1λ3−3λ2λ3, P− 312 =P+ 312 −2k1k2k3(λ1−λ2)λ2 3+ 3λ1λ2+ 4λ1λ3+ 4λ2λ3, P− 313 =−P+ 313 + 2k2k3λ2λ2 1−λ2 3+λ1λ2+ 3λ1λ3−4λ2λ3, P− 314 =P+ 314 + 2k1k3λ1λ2 2−λ2 3+λ1λ2−4λ1λ3+ 3λ2λ3, P− 412 =P+ 412 + 2k1k2(λ1−λ2)λ2 3−3λ1λ2+λ1λ3+λ2λ3, P− 413 =P+ 413 + 2k1k3(λ1−λ3)λ2 2+λ1λ2−3λ1λ3+λ2λ3, P− 414 =−P+ 414 −2k2k3(λ2−λ3)λ2 1+λ1λ2+λ1λ3−3λ2λ3. Since λ1= 0, we can consider the orthogonal basis ˆei=1 λ1eiand assume that λ1= 1 for the rest of our calculations, just working in the homothetic class of the 244 6.2 Half-harmonic Weyl curvature on homogeneous four-manifolds P314 =−(2A2C+AD + 2C)λ2 1+Cλ2 2−(AD −C)λ1λ2+b(2AD + 9C)λ1 −b(10AD + 3C)λ2+ 2C(3b2−C2−D2), P412 = 2A(A2+ 1)(λ3 1+λ3 2−λ2 1λ2−λ1λ2 2) + (2AC2−AD2−3CD)λ1 −(AC2−2AD2−3CD)λ2, P413 = (2A2C−AD + 2C)λ2 1−A2Cλ2 2−A(AC +D)λ1λ2+b(AD + 4C)λ1 −3AbDλ2+ 2C(b2+C2+D2), P414 =−A2Dλ2 1+ (2A2D+AC + 2D)λ2 2−A(AD −C)λ1λ2+ 3AbCλ1 −b(AC −4D)λ2+ 2D(b2+C2+D2). Since λ1λ2= 0, we consider the orthonormal basis ˆei=1 λ1eiand work in the homothetic class of the initial metric. This way, we can assume that λ1= 1 in what follows. The divergence δW +vanishes if and only if the structure constants in equation (6.5) satisfy the system of polynomial equations {Pijk = 0}, where Pijk ∈R[λ2, A, b, C, D]. Let I1⊂R[λ2, A, b, C, D]be the ideal generated by the polynomials Pijk. We compute a Gröbner basis G1of I1with respect to the lexicographical order and we see that one of the 39 polynomials in it is g1=D5C2+D249D2+ 1649D2+ 144193600D4+ 16560D2+ 2187. It follows immediately that D= 0. Now we compute a Gröbner basis G2of the ideal I2generated by the polynomials G1∪{D} ⊂ R[λ2, A, b, C, D]with respect to the lexicographical order, obtaining that the polynomial g2=CA2+ (b+ 1)2+C2 belongs to G2. Therefore, C= 0, and the polynomial P312 is reduced to P312 = 2 A2+ 1(λ2−1)2(λ2+ 1) . This leads to two different possibilities depending on whether λ2= 1 or λ2=−1. If λ2= 1, then the left-invariant metric is locally conformally flat. Moreover, it is flat or locally isometric to a product R×N(c), where N(c)is a three-dimensional manifold of constant sectional curvature. If λ2=−1, the polynomial P114 is reduced to P114 = 8 b2−A2−1, so b=±√A2+ 1. In this situation, the corresponding left-invariant metric is Einstein (thus locally symmetric [88]) if A= 0 and locally isometric to a product M1(c1)×M2(c2)of two surfaces of constant curvature c2 1=c2 2otherwise. The Weyl tensor is divergence-free in all the cases above, which shows that there are no non-trivial examples in this case. 251 Homogeneous four-manifolds with half-harmonic Weyl curvature Remark 6.10.Proceeding in a completely analogous way, one gets that the metrics with half-harmonic Weyl conformal tensor corresponding to the condition δW −= 0 are again the ones described just above. 252 Chapter 7 Three-dimensional Riemannian homogeneous structures In this chapter we will devote ourselves to the study of homogeneous structures and give a complete classification of the homogeneous structures on non-symmetric three-dimensional Riemannian Lie groups. We will see that one such group admits a non-canonical homogenous structure if and only if its isometry group is fourdimensional. The results in this chapter are contained in the work [38]. Before we start, we will make a short introduction to the world of homogeneous structures. 7.1 Homogeneous structures The characterization of Riemannian locally symmetric spaces as those whose curvature is parallel with respect to the Levi-Civita connection was originally given by Cartan. Ambrose and Singer extended this characterization to homogeneous Riemannian manifolds showing that a connected, complete and simply connected ndimensional Riemannian manifold (M, g)is homogeneous if and only if there exists a(1,2)-tensor field Ton Msuch that e ∇g= 0,e ∇R= 0,e ∇T= 0, (7.1) where e ∇is the Ambrose-Singer connection given by e ∇=∇−T,∇is the Levi-Civita connection of the metric g, and Rdenotes the Riemannian curvature tensor for which we adopt the sign convention R(X, Y ) = ∇[X,Y ]−[∇X,∇Y](see [3]). The tensor field Tis said to be a homogeneous structure on M. We will also denote by Tthe associated (0,3)-tensor field given by T(X, Y, Z) = g(T(X, Y ), Z). Conditions (7.1) were further investigated by Tricerri and Vanhecke in [130], where they considered the space T(V)of such tensor fields on a vector space (V,⟨·,·⟩)and decomposed it into three irreducible components under the action of the orthogonal 253 Three-dimensional Riemannian homogeneous structures group as T(V) = T1(V)⊕T2(V)⊕T3(V). The subspaces of such decomposition are T1(V) = {T∈ T(V): T(x, y, z) = ⟨x, y⟩φ(z)−⟨x, z⟩φ(y)}, T2(V) = {T∈ T(V): c12(T) = 0, σx,y,zT(x, y, z) = 0}, T3(V) = {T∈ T(V): T(x, y, z) + T(y, x, z) = 0}, where φ∈ V∗,σx,y,z is the cyclic sum with respect to x, y, z and c12(T)denotes the contraction c12(T)(z) = X i T(ei, ei, z) for an arbitrary orthonormal basis {ei}of V. The projections of a homogeneous structure Ton each of these subspaces are given by p1(T)(x, y, z) = 1 n−1⟨x, y⟩c12(T)(z)−1 n−1⟨x, z⟩c12(T)(y), p3(T)(x, y, z) = 1 3σx,y,zT(x, y, z), p2(T)(x, y, z)=(T−p1(T)−p3(T)) (x, y, z). (7.2) Homogeneous manifolds admitting a homogeneous structure in one of the eight different classes induced by the decomposition above have been extensively studied in the literature. It was shown in [130] that naturally reductive spaces correspond to non-vanishing homogeneous structures of type T3and that a Riemannian manifold admits a non-vanishing structure of type T1if and only if it is locally isometric to the real hyperbolic space. The latter also holds true for homogeneous structures of type T1⊕ T3,T /∈ T1and T /∈ T3, in dimension greater than three, as shown in [117]. Riemannian manifolds of dimension less than or equal to four that admit a homogeneous structure of type T2were described in [93] (see also [34]). Homogeneous structures in the class T1⊕T2in dimension less than or equal to four were described in [73], and those in this class whose fundamental one-form is closed were investigated in [118]. It was shown in [93] that a three-dimensional non-symmetric space admitting a homogeneous structure of type T3also admits a T2-structure. In dimension two, the decomposition above reduces to T(V) = T1(V). As a consequence, a surface admits a non-zero homogeneous structure if and only if it is isometric to the hyperbolic plane. Dimension three is particularly relevant to the study of homogeneous spaces. First, it is the lowest possible dimension admitting locally homogeneous metrics which are not locally symmetric and, secondly, any three-dimensional homogeneous manifold is either symmetric or locally isometric to a Lie group endowed with a left-invariant metric [126]. 254 7.2 Riemannian homogeneous structures in dimension three The special case in which (M, g)is a Lie group Gequipped with a left-invariant metric ⟨·,·⟩is of special interest for our purposes. Let T∇be the canonical homogeneous structure defined by 2⟨T∇(X, Y ), Z⟩=⟨[X, Y ], Z⟩−⟨[Y, Z], X⟩+⟨[Z, X], Y ⟩, for left-invariant vector fields X, Y and Z. Then the corresponding Ambrose-Singer connection e ∇=∇−T∇satisfies e ∇XY= 0 for any two left-invariant vector fields. This structure is equivalent to the description G=G/{e}, which corresponds to the action G×G→G. 7.2 Riemannian homogeneous structures in dimension three On the basis of the above outlined, in this section we will clarify the classification of the Riemannian homogeneous structures in dimension three, describing all the possible ones in the non-symmetric case. The following result characterizes the nonsymmetric Lie groups admitting more than one homogeneous structure. Theorem 7.1. A non-symmetric simply connected three-dimensional Riemannian Lie group admits a homogeneous structure different from the canonical one if and only if it admits a naturally reductive homogeneous structure. Moreover, in such a case, it admits exactly a one-parameter family of homogeneous structures. The explicit description of all the homogeneous structures on non-symmetric Lie groups is given in Theorem 7.2 and Theorem 7.3. Recall that a three-dimensional complete and simply connected manifold is naturally reductive if and only if it admits a non-vanishing homogeneous structure of type T3. In this case (M, g)is a real space form R3,S3or H3, or it is isometric either to the special unitary group SU(2), or to the universal covering of f SL(2,R)or to the 3-dimensional Heisenberg group H3, endowed with a suitable left-invariant metric described in terms of the Lie algebras (up to rotations) as H3: [e1, e2] = λe3, λ = 0, SU(2) : [e1, e2] = µe3,[e2, e3] = λe1,[e3, e1] = λe2, λ µ > 0, f SL(2,R) : [e1, e2] = µe3,[e2, e3] = λe1,[e3, e1] = λe2, λ µ < 0, (7.3) where {e1, e2, e3}is an orthonormal basis (see [130]). In this way, (M, g)is naturally reductive if and only if it is isometric to a Lie group endowed with a left-invariant metric whose isometry group is at least fourdimensional. 255 Three-dimensional Riemannian homogeneous structures Theorem 7.1 is thus connected to the following theorem by Meeks and Perez (see [103]): a simply connected, three-dimensional Lie group with a left-invariant metric (G1,⟨·,·⟩1)is isometric to a second Lie group (G2,⟨·,·⟩2)which is not isomorphic to G1if and only if its isometry group has dimension at least four. 7.2.1 Summary of results We study the unimodular and non-unimodular cases separately. The unimodular case is dealt with in Section 7.2.2, and the non-unimodular case is considered in Section 7.2.3. We will see that the Riemannian homogeneous structures on a non-symmetric three-dimensional Lie group Gequipped with a left-invariant metric are given as follows, from where the proof of Theorem 7.1 is obtained at once. Unimodular Lie groups The left-invariant Riemannian metrics ⟨·,·⟩ on three-dimensional unimodular Lie groups Gwere described by Milnor (see [104]) in terms of three structure constants (λ1, λ2, λ3), so that the Lie algebra becomes [e1, e2] = λ3e3,[e1, e3] = −λ2e2,[e2, e3] = λ1e1, for an orthonormal basis {e1, e2, e3}. It now follows that (G, ⟨·,·⟩)is symmetric if and only if λ1= 0 and λ2=λ3(up to rotations), in which case it is flat, or λ1=λ2=λ3= 0, and the sectional curvature is constant and positive. We focus on the non-symmetric situation and we have the following result. Theorem 7.2. Let (G, ⟨·,·⟩)be a unimodular Lie group equipped with a non-symmetric left-invariant Riemannian metric. Then there are two mutually excluding cases. (i) The three structure constants λ1, λ2, λ3are different and the only homogeneous structure is the canonical one, which is given by T∇=−(λ1−λ2−λ3)e1⊗(e2∧e3)−(λ1−λ2+λ3)e2⊗(e1∧e3) +(λ1+λ2−λ3)e3⊗(e1∧e2). The canonical homogeneous structure is of type T2if λ1+λ2+λ3= 0 (see also [73]) and it is of type T2⊕T3otherwise. (ii) Up to a rotation, the structure constants λ1=λ2=λ3,λ3= 0 and there exists a one-parameter family of homogeneous structures T=λ3e1⊗(e2∧e3)−λ3e2⊗(e1∧e3)+2κe3⊗(e1∧e2), κ ∈R, 256 7.2 Riemannian homogeneous structures in dimension three which corresponds to the canonical structure for κ=1 2(2λ1−λ3). Moreover, it is of type T2if κ=−λ3, of type T3if κ=1 2λ3, and of type T2⊕T3otherwise. The unimodular Lie groups in Theorem 7.2-(ii) correspond to SU(2),f SL(2,R) and H3with left-invariant metric as in (7.3), and these include the homogeneous structures on Berger spheres previously considered in [75]. Non-unimodular Lie groups Non-unimodular Riemannian Lie groups (G, ⟨·,·⟩)are semi-direct extensions R2⋊R of the Abelian group. It was shown in [104] that there exists an orthonormal basis {e1, e2, e3}so that [e1, e2] = αe2+βe3,[e1, e3] = γe2+δe3,[e2, e3] = 0, where the trace of the endomorphism determining the semi-direct extension satisfies α+δ= 0. Moreover, one may rotate the orthonormal basis {e2, e3}of the unimodular kernel to assume that their images by the endomorphism are orthogonal, i.e., αγ +βδ = 0. The Riemannian Lie group (G, ⟨·,·⟩)is symmetric if and only if β=δ=γ= 0 (up to the isometry e27→ e3), and so it is isometric to R×H2(−α2), or if α=δ= 0 and γ=−β, in which case it is a space of constant sectional curvature H3(−δ2). In this setting, we have the following result. Theorem 7.3. Let (G, ⟨·,·⟩)be a non-unimodular Lie group equipped with a leftinvariant non-symmetric Riemannian metric. Then there are two mutually excluding cases. (i) If δ=γ= 0,αβ = 0, then the homogeneous structures are given by one of the following possibilities. (i.a) The one-parameter family T=βe1⊗(e2∧e3)−βe2⊗(e1∧e3)+2κe3⊗(e1∧e2), κ ∈R. In this case the homogeneous structure is of type T2if κ=−β, of type T3if κ=1 2β, and of type T2⊕T3otherwise. (i.b) The canonical homogeneous structure T∇=βe1⊗(e2∧e3)−2αe2⊗(e1∧e2)−βe2⊗(e1∧e3)−βe3⊗(e1∧e2), which is of the generic type T1⊕T2⊕T3. 257 Three-dimensional Riemannian homogeneous structures (ii) If δα = 0,β=−αγ δand α=δ, then the only homogeneous structure is the canonical one, which is given by T∇=−(α+δ)γ δe1⊗(e2∧e3)−2αe2⊗(e1∧e2) + (α−δ)γ δe2⊗(e1∧e3) +(α−δ)γ δe3⊗(e1∧e2)−2δe3⊗(e1∧e3), and is of type T1⊕T2if γ= 0, and T1⊕T2⊕T3otherwise. The non-unimodular Lie groups given in Theorem 7.3 are semi-direct extensions R2⋊ R of the Abelian Lie group determined by an endomorphism −ad(e1). Assertion (i) in Theorem 7.3 corresponds to the special situation where det ad(e1) = 0, and they are isometric (although not isomorphically isometric) to a left-invariant metric on f SL(2,R)as in (7.3), which corresponds to Theorem 7.2-(ii) (cf. [103, 130]). We would like to emphasize that isometries between Riemannian Lie groups need not preserve the Lie group structure, since they are not necessarily realized by group isomorphisms, as evidenced in the above-mentioned situation. On the other hand, the Lie groups in Theorem 7.3-(ii) correspond to the generic situation, where one can always specify the orthonormal basis {e2, e3}so that it is given by eigenvectors of the self-adjoint part of ad(e1)(cf. [104]). Non-symmetric simply connected homogeneous three-dimensional Riemannian manifolds with four-dimensional isometry group are isometric to the unitary group SU(2), the universal cover of f SL(2,R), or the Heisenberg group with the special metrics (7.3). It follows from the description of homogeneous structures in Theorem 7.2 and Theorem 7.3 that (see also [130]) a non-symmetric three-dimensional Riemannian Lie group admits a homogeneous structure different from the canonical one if and only if its isometry group is four-dimensional Remark 7.4.A more conceptual proof of this last statement can be summarized as follows. For any three-dimensional Lie groups (G1,⟨·,·⟩1)and (G2,⟨·,·⟩2)equipped with a left-invariant Riemannian metric, it follows from Theorems 7.2 and 7.3 that the infinitesimal models associated to their canonical homogeneous structures are isomorphic if and only if the Lie groups (G1,⟨·,·⟩1)and (G2,⟨·,·⟩2)are isomorphically isometric (see [34]). Besides, any non-symmetric homogeneous three-manifold with four-dimensional isometry group admits more than one homogeneous structure. It follows from the works in [103,126] that a homogeneous three-manifold with threedimensional isometry group is isometric to a unique Riemannian Lie group, in which case any homogeneous structure is isomorphic to the canonical one. 258 7.2 Riemannian homogeneous structures in dimension three 7.2.2 Homogeneous structures on non-symmetric unimodular Lie groups Following [104], if gis unimodular then there exists an orthonormal basis {e1, e2, e3} of gsuch that [e1, e2] = λ3e3,[e1, e3] = −λ2e2,[e2, e3] = λ1e1. Let Tbe a (0,3)-tensor field so that the connection e ∇=∇−Tmakes the metric tensor parallel, i.e., Txyz +Txzy = 0 for x,y,z∈g. Denoting by {e1, e2, e3}the dual basis of {e1, e2, e3}, then the tensor field Tcan be written as T= 2 X iX j<k Tijkei⊗(ej∧ek). Therefore, the non-zero components of the connection e ∇=∇−Tare given by e ∇112 =−T112,e ∇223 =−T223,e ∇123 =−1 2(λ1−λ2−λ3)−T123, e ∇113 =−T113,e ∇313 =−T313,e ∇213 =−1 2(λ1−λ2+λ3)−T213, e ∇212 =−T212,e ∇323 =−T323,e ∇312 =1 2(λ1+λ2−λ3)−T312, (7.4) while the (0,4)-curvature tensor field is determined by R1212 =1 4(λ1−λ2)2−3λ2 3+ 2(λ1+λ2)λ3, R1313 =1 4(λ1−λ3)2−3λ2 2+ 2(λ1+λ3)λ2, R2323 =1 4(λ2−λ3)2−3λ2 1+ 2(λ2+λ3)λ1. (7.5) Let Rikjℓ;r= (e ∇erR)(ei, ej, ek, eℓ). A straightforward calculation involving Equations (7.4) and (7.5) shows that the condition e ∇R= 0 in Equation (7.1) is given by 259 Three-dimensional Riemannian homogeneous structures 2R1213;1 = (λ1−λ2−λ3)(λ2−λ3)(λ1−λ2−λ3+ 2T123) = 0, R1213;2 = (λ1−λ2−λ3)(λ2−λ3)T223 = 0, R1213;3 = (λ1−λ2−λ3)(λ2−λ3)T323 = 0, R1223;1 = (λ1−λ2+λ3)(λ1−λ3)T113 = 0, 2R1223;2 = (λ1−λ2+λ3)(λ1−λ3)(λ1−λ2+λ3+ 2T213)=0, R1223;3 = (λ1−λ2+λ3)(λ1−λ3)T313 = 0, −R1323;1 = (λ1+λ2−λ3)(λ1−λ2)T112 = 0, −R1323;2 = (λ1+λ2−λ3)(λ1−λ2)T212 = 0, 2R1323;3 = (λ1+λ2−λ3)(λ1−λ2)(λ1+λ2−λ3−2T312) = 0. (7.6) From here, depending on the eigenvalues λi, we are led to the following two possibilities. Case of three different eigenvalues In this case, if λ1−λ2−λ3,λ1−λ2+λ3and λ1+λ2−λ3do not vanish, then Equations (7.4) and (7.6) clearly imply that e ∇XY= 0 for all left-invariant vector fields. Therefore, the only homogeneous structure is the canonical one given by Txy=∇xy, for x,y∈g, i.e., T=−(λ1−λ2−λ3)e1⊗(e2∧e3)−(λ1−λ2+λ3)e2⊗(e1∧e3) + (λ1+λ2−λ3)e3⊗(e1∧e2). (7.7) Next we show that the same holds if any of λ1−λ2−λ3,λ1−λ2+λ3and λ1+λ2−λ3vanishes. Suppose that λ3=λ1−λ2(the other two cases are obtained in a completely analogous way). Then, Equations (7.6) implies T112 =T113 =T212 =T313 = 0, T213 =−λ1+λ2, T312 =λ2.(7.8) Let Tijk;r= (e ∇erT)(ei, ej, ek). A straightforward calculation using Equations (7.4) and (7.8) shows that the condition e ∇T= 0 in Equation (7.1) reduces to T313;r=−T212;r= (λ1−2λ2)Tr23 = 0, T223;r=Tr23T323 = 0, T323;r=−Tr23T223 = 0, 260 7.3 Self-dual and anti-self-dual homogeneous structures (i) [e1, e2] = e3,[e1, e4] = αe1,[e2, e4] = αe2,[e3, e4] = 2αe3,α= 0,±1 2. In this case, T=−2αe1⊗(e1∧e4) + e1⊗(e2∧e3)−e2⊗(e1∧e3) −2αe2⊗(e2∧e4)−(2α−1)e3⊗(e1∧e2)−4αe3⊗(e3∧e4) is of type T1⊕T+ 2if α=3 2, or of type T1⊕T+ 2⊕T3otherwise. (ii) [e1, e2] = e3,[e1, e4] = e1,[e2, e4] = −1 2e2,[e3, e4] = 1 2e3. In this case, T=−e2⊗(e1∧e3) + e2⊗(e2∧e4)−e3⊗(e1∧e2)−e3⊗(e3∧e4) is of type T+ 2. There are semi-direct extensions of the Abelian group R3which admit self-dual homogeneous structures. Considering the Lie algebra [e1, e4] = e1,[e2, e4] = fe2+he3,[e3, e4] = −he2+fe3, f = 1, where {e1, e2, e3, e4}is an orthonormal basis, one can check that the homogeneous structures T=−2e1⊗(e1∧e4) + 2(f−1)e1⊗(e2∧e3) −2fe2⊗(e2∧e4)−2fe3⊗(e3∧e4) are self-dual and never correspond to the canonical homogeneous structure. 267 Conclusions and open problems The main achievements of this work can be outlined as follows. C.1 We completed the classification of four-dimensional locally conformally flat Kähler, para-Kähler and null-Kähler structures. C.2 We determined all the left-invariant four-dimensional para-Kähler Lie groups up to automorphisms preserving the symplectic structure, showing that all the locally conformally flat structures are realizable as left-invariant Kähler or para-Kähler structures on Lie groups. C.3 We gave a complete description of all the left-invariant Ricci solitons on fourdimensional Lorentzian Lie groups. C.4 We determined all the Riemannian algebraic Bach solitons, showing that they are algebraic Ricci solitons or belong to one of two exceptional families. C.5 We classified the Riemannian homogeneous four-dimensional manifolds with half-harmonic Weyl curvature. C.6 We showed that a non-symmetric three-dimensional homogeneous manifold admits more than one homogeneous structure if and only if its isometry group has dimension four. A number of open problems naturally arise as a consequence of our work. P.1 The complete classification of four-dimensional Bochner-flat para-Kähler structures of non-constant scalar curvature. It was shown in [67] that any Bochner flat para-Kähler surface of constant scalar curvature is locally a para-complex space form or a locally conformally flat para-Kähler surface as described in Theorem 2.1. The problem of non-constant scalar curvature remains open. Such Bochner flat para-Kähler surfaces are locally isometric to a cotangent bundle with a modified Riemannian extension but a precise parametrization of such structures is still under consideration. 269 Three-dimensional Riemannian homogeneous structures P.2 Algebraic Lorentzian Ricci solitons. Algebraic Ricci solitons on Lorentzian Lie groups are well-understood in dimension three [14]. One expects to be able to solve the four-dimensional case by following the strategy developed in Chapter 5, although the calculations seem to be much more involved. P.3 The existence of non-trivial left-invariant Bach solitons. The existence of homogeneous gradient Bach solitons and algebraic Bach solitons on four-dimensional Riemannian Lie groups was discussed in Chapter 5. While there are no non-trivial Ricci solitons on four-dimensional Lie groups with a left-invariant soliton vector field, it is an open problem whether the same statement holds true in the case of Bach solitons. P.4 Algebraic T-solitons. It was shown by Arroyo and Lafuente [9] that any Riemannian expanding homogeneous Ricci soliton is homothetic to an algebraic Ricci soliton in dimension four. Hence, four-dimensional homogeneous Ricci solitons are either symmetric or algebraic. The situation seems to be much more complicated for other geometric flows where it is not clear whether any non-symmetric soliton is necessarily algebraic. 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