Left-invariant distributions diffeomorphic to flat distributions
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This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Left-invariant distributions diffeomorphic to flat distributions © 2024 the Authors Published version Nicolussi Golo, Sebastiano; Ottazzi, Alessandro Nicolussi Golo, S., & Ottazzi, A. (2024). Left-invariant distributions diffeomorphic to flat distributions. Geometriae Dedicata, 218(2), Article 56. https://doi.org/10.1007/s10711-02400905-3 2024
Geometriae Dedicata (2024) 218:56 https://doi.org/10.1007/s10711-024-00905-3 ORIGINAL PAPER Left-invariant distributions diffeomorphic to flat distributions Sebastiano Nicolussi Golo1 ·Alessandro Ottazzi2 Received: 14 July 2022 / Accepted: 14 February 2024 / Published online: 13 March 2024 © The Author(s) 2024 Abstract For a stratified group G, we construct a class of Lie groups endowed with a left-invariant distribution locally diffeomorphic to the flat distribution of G.Viceversa,weshowthatallLie groups with a left-invariant distribution that is locally diffeomorphic to the flat distribution of Gbelong to the class we constructed, if the Lie algebra of Ghas finite Tanaka prolongation. Keywords Flat distributions ·Tanaka prolongation ·Stratified Lie groups ·Contact structures ·Quasi-conformal maps Mathematics Subject Classification (2010) 30L10 ·22E25 ·53C30 Contents 1 Introduction ............................................... 2 2 Notation and preliminaries ........................................ 4 2.1 Polarizations and Tanaka prolongations ............................... 4 2.2 The groups Pand Qand their quotient M............................. 5 2.3 Polarizations on G,Pand M.................................... 7 3 Distribution-preserving diffeomorphisms of Mwhen Gis rigid .................... 8 4 Modifications of stratified groups ....................................11 5 Examples .................................................13 5.1 Modifications of the Heisenberg group ...............................13 Sebastiano Nicolussi Golo has been partially supported by the European Unions Seventh Framework Programme, Marie Curie Actions-Initial Training Network, under Grant Agreement No. 607643, “Metric Analysis For Emergent Technologies (MAnET)”, and by the EPSRC Grant "Sub-Elliptic Harmonic Analysis" (EP/P002447/1), and by University of Padova STARS Project "Sub-Riemannian Geometry and Geometric Measure Theory Issues: Old and New". Alessandro Ottazzi has been partially supported by the ARC Discovery grant DP170103025. Data sharing not applicable to this article as no datasets were generated or analysed during the current study. BAlessandro Ottazzi [email protected] Sebastiano Nicolussi Golo [email protected] 1Department of Mathematics and Statistics, University of Jyväskylä, 40014 Jyväskylä, Finland 2School of Mathematics and Statistics, University of New South Wales (UNSW), Sydney 2052, Australia 123
56 Page 2 of 22 Geometriae Dedicata (2024) 218 :56 5.1.1 Rigid motions of the plane as a modification of the Heisenberg group ...........17 5.2 Modifications of the free nilpotent Lie group F24 .........................17 5.3 Modifications of ultra-rigid stratified groups ............................20 References ..................................................21 1 Introduction In this article, we consider the following question: given a stratified group G, we wish to characterise those polarised Lie groups that are equivalent to G.Hereapolarisation on a Lie group is the choice of a left-invariant and bracket-generating subbundle of the tangent bundle (cfr. [11]), and two polarized Lie groups are equivalent if there is a locally defined distribution-preserving diffeomorphism between them. Stratified groups carry a canonical polarisation. Given a stratified group, we will construct a class of polarised Lie groups that are equivalent to G, which we will call modifications of G. The key tool for our construction will be Tanaka prolongation theory. Before diving in the technical details of our main results, we first provide some framework. Theproblemunderstudyisrelevantindifferent areas, such as Tanaka prolongation theory,CR geometry, sub-Riemannian geometry, and control theory. In the setting of Tanaka’s theory, it is known that the infinitesimal automorphisms of the polarisation associated to a stratified Lie algebraareencodedbyitsfullTanakaprolongation(see,e.g.,[13,15,18]).IftheLiealgebrais not stratified, however, all we can conclude is that every infinitesimal automorphism induces an infinitesimal automorphism on its stratified symbol. In this paper, we construct classes of polarised Lie algebras that are not stratified, but that have the same space of infinitesimal automorphisms as their stratified symbol. Our study has potential applications to geometric control theory. Given a nonholonomic control system, the motion planning problem consists in finding a curve tangent to the polarisationthat connectstwo givenpointsin the ambientspace.Nilpotent Lie groupsarethe widest class of nonholonomic systems for which an exact solution to the motion planning problem is known, see [6]. Distribution-preserving diffeomorphisms are equivalences of motion planning problems. Thus, our method detects classes of non-nilpotent nonholonomic systems that are equivalent to nilpotent ones. Furthermore, our findings have consequences in metric geometry. On a polarised Lie group, one may define a left-invariant sub-Riemannian distance. In metric geometry, it is natural to study the equivalence of metric spaces up to isometries, bi-Lipschitz mappings, conformal and quasiconformal mappings. For example, if two stratified groups are (locally) quasiconformal, then their Lie algebras are isomorphic [14]. If two nilpotent Lie groups are isometric, then they are isomorphic [7,9]. It is an open question to determine whether two nilpotent Lie groups that are globally bi-Lipschitz to one another are indeed isomorphic. In [4], the authors study the Lie groups that can be made isometric to a given nilpotent Lie group, endowed with a left-invariant distance. (See also [5] for the Riemannian case.) In this sense, our work follows [4], because distribution-preserving diffeomorphisms are locally bi-Lipschitz. In sub-Riemannian geometry, one of the major open problems is to determine whether the conclusions of Sard Theorem hold for the endpoint map, which is a canonical map from an infinite dimensional path space to the underlying finite dimensional manifold. The set of critical values for the endpoint map is also known as abnormal set, being the set of endpoints of abnormal extremals leaving the base point. In the context of Lie groups, perhaps the most general positive results have been proved in [11]. Here the authors prove 123
Geometriae Dedicata (2024) 218 :56 Page 3 of 22 56 that the abnormal set has measure zero in the case of 2-step stratified groups and several other examples. This property for the abnormal set is preserved by distribution-preserving diffeomorphisms between sub-Riemannian manifolds. It then comes out from our results that every modification of a stratified group satisfies the Sard Theorem, if the stratified group does. Now we will present our main results in detail. Recall that a Tanaka prolongation of a stratified Lie algebra gthrough a Lie subalgebra g0of the Lie algebra of derivations of g that preserve the stratification is the maximal nondegenerate graded Lie algebra that contains g+g0, where nondegenerate means that the adjoint action of any element of positive weight on gis nontrivial.1When g0is chosen as the whole set of strata preserving derivations, we obtain the full Tanaka prolongation. When the prolongation algebra is finite dimensional, we obtain a graded Lie algebra p=g⊕qand we say that gand any Lie group Gwith Lie algebra gare rigid.Thetermof finite type is also common in the literature to denote Lie algebras with finite Tanaka prolongation. Modifications of gare then defined to be subalgebras sof pof the same dimension of gthat are transversal to q. It turns out that there is a linear map σ:g→qof which the modification is the graph. If g−1is the first layer of g, then the set {v+σ(v) :v∈g−1}defines a polarisation on a Lie group whose Lie algebra is the modification s. Our first main result draws the connection between modifications of Gand Lie groups that are equivalent to G. Theorem A Every modification of a stratified Lie group G is equivalent to G. Viceversa, if G is rigid, then every polarised Lie group that is equivalent to G is one of its modification. Theorem Ais restated and proven in Theorems 4.2 and 4.4. A key tool in the study of local distribution-preserving diffeomorphisms is the quotient manifold M=P/Q,whereP and Q<Pare the Lie groups with Lie algebras pand qrespectively. The polarisation of G induces a polarisation Mon Mand Gembeds in Mas an open subset, see Proposition 2.9. Moreover, if Sis a modification of G, then an open neighborhood of the identity in Scan be also embedded into M, see Lemma 4.1. The composition of such embeddings induce a local distribution-preserving diffeomorphism between Gand S. If the group Gis rigid, i.e., its full Tanaka prolongation is finite dimensional, then all distribution-preserving diffeomorphisms between Gand Sarise in this way, see Theorem 4.4. The rigid case is particularly favorable because all distribution-preserving diffeomorphisms of Gare induced by affine maps on P,see(3) at page 9. We can express this rigidity in terms of local distribution-preserving diffeomorphisms of M. More precisely, we will prove in Theorem 3.3 the following statement: Theorem B Suppose G is rigid and let M =P/Q be the manifold described above, where the Lie algebra of P is the full tanaka prolongation of g.LetU ⊂M be open and connected and f :U→f(U)⊂M a smooth map with d f (M)⊂M. Suppose that there exists x0∈U such that d f (x0)is non-singular. Then there exists a unique distribution-preserving diffeomorphism g :M→M such that g|U=f. We will also prove that, in the hypothesis of Theorem B, the connected component of the identity in the group of distribution-preserving diffeomorphisms of Mis isomorphic to Q, see Theorem 3.6. 1Recallthata grading ofaLiealgebra isavectorspacedecompositiong=i∈Zgisuchthat[gi,gj]⊂gi+j for all i,j∈Z. A grading is a stratification if g=i≤−1giand [g−1,gj]=gj−1for all j<0. 123
56 Page 4 of 22 Geometriae Dedicata (2024) 218 :56 It remains open whether the second part of Theorem Aholds true without asking that the full Tanaka prolongation is finite. While we cannot prove the theorem in this generality, examplessuggestthatitmaybetrue.Moreprecisely,inSect.5.1,weshowthatallthreedimensional sub-Riemannian structures are modifications of the Heisenberg group with respect to a suitable finite dimensional Tanaka prolongation, even though the full prolongation of the Heisenberg Lie algebra is infinite dimensional, see Theorem 5.1. This justifies the following conjecture. Conjecture Suppose that G is a stratified Lie group and that S is a polarised Lie group that is equivalent to G. Then there is a finite Tanaka prolongation of Lie(G)in which Lie(S)is a modification of Lie(G). In Sect. 5.2 we explicitly compute some modifications of the free nilpotent Lie group with two generators and step four, F24. It comes out that one may construct examples of non-nilpotent Lie groups that are equivalent to F24. We also find a nilpotent, non-stratified, polarised Lie group that is equivalent to F24 via a global distribution-preserving diffeomorphism, see Theorem 5.5. Finally, in Sect. 5.3, we study all the modifications of an ultra-rigid stratified group, that is, a stratified group whose only strata-preserving derivation is the infinitesimal generator of dilations. It turns out that such modifications are all solvable and the only nilpotent one is the stratified group itself, see Theorem 5.9. The paper is organized as follows. In Sect.2, we fix the notation and establish the framework in which we will be working. We consider stratified algebras and their Tanaka prolongations, we define the corresponding Lie groups and fix a polarisation on them. In Sect.3, we study distribution-preserving diffeomorphisms of Mas affine maps of Pand prove Theorem B. In Sect.4, we define the modifications of a stratified algebra and those of a stratified group, proving Theorem A. Finally, we apply our modification technic to a number of examples in Sect.5. 2 Notation and preliminaries 2.1 Polarizations and Tanaka prolongations Given a connected, smooth manifold M,apolarisation of Mis the choice of a subbundle M of the tangent bundle TMthat is bracket generating, i.e., with the property that the sections of Mbracket generate all the sections of TM. Given two polarised manifolds (M,M)and (N,N),adistribution-preserving diffeomorphism between Mand Nis a diffeomorphism f:M→Nsuch that f∗(M)=N.Wedenoteby(TM)the space of vector fields on M. A vector field V∈(TM)on a polarised manifold (M,M)is a contact vector field if its flow is made of distribution-preserving diffeomorphisms. For a Lie group S,weshall always consider left-invariant polarisations S. The pair (S,S)is called a polarised group. The identity element will be denoted by eS,orsimplyeif no confusion arises. We denote by Gastratified group, that is, a connected and simply connected Lie group whose Lie algebra decomposes as g=−1 i=−sgi, with [g−1,gj]=gj−1for every −s+1≤j≤−1. On a stratified group we will always consider the left-invariant polarisation Gfor which (G)eG=g−1. In a stratified group Gwe consider the strata preserving derivations Der(g):= {u∈End(g):u(g−1)⊂g−1, and u[X,Y]=[u(X), Y]+[X,u(Y)]∀X,Y∈g}. 123
Geometriae Dedicata (2024) 218 :56 Page 5 of 22 56 Given a subalgebra g0of Der(g),wedefinetheTanaka prolongation of gthrough g0as the (possibly infinite) maximal nondegenerate graded Lie algebra Prol(g,g0)=k≥−sgk which contains g⊕g0. When g0=Der(g), we call Prol(g,g0)the full Tanaka prolongation of g. It is not difficult to see that the latter contains all prolongations. We say that g,orG,is rigid if the full Tanaka prolongation has finite dimension. When it is clear from the context and the prolongation under consideration is finite dimensional, we shall denote Prol(g,g0) by p, the nonnegative part k≥0gkby q, and the positive part k>0gkby p+.See[13,15, 18] for further details on Tanaka prolongation. 2.2 The groups Pand Qand their quotient M In the following, we establish a number of properties of the Lie groups that correspond to the Lie algebras introduced above. Let ¯ Pbe the connected and simply connected Lie group whose Lie algebra is a finite dimensional Tanaka prolongation pof a stratified Lie algebra g. Let ¯ Qbe the connected subgroup of ¯ Pwhose Lie algebra is q. The set {δλ:λ>0}of mappings on pdefined by δλ(X)=λiXfor X∈giis a one parameter family of automorphisms of p. By abuse of notation, we write δλfor the corresponding automorphisms of the group ¯ P. Such maps exist because ¯ Pis simply connected. Lemma 2.1 Denote by expP:p→¯ P the exponential map of ¯ P. Then expPis injective on gand on k≥1gk. Proof Let v,w ∈gsuch that expP(v) =expP(w).Sincev,w ∈g, then limλ→∞ δλv= limλ→∞ δλw=0. Let λ≥1 be such that both δλ(v) and δλ(w) belong to a neighborhood U of 0 in pon which the exponential map expPis injective. Then expP(δλv) =δλ(expP(v)) = δλ(expP(w)) =expP(δλw). By the injectivity of expPon U,wehaveδλv=δλw.Sinceδλ is a linear isomorphism, we conclude that v=w. A similar argument proves that expPis injective on k≥1gk. By Lemma 2.1, the canonical immersion G→¯ Pinduced by g→pis injective. We are going to show that Gis closed in ¯ P. We prove two lemmas first. Lemma 2.2 The intersection of G with ¯ Qistrivial. Proof Since δλ(g)=gand δλ(q)=q,thenδλ(G)=Gand δλ(¯ Q)=¯ Q,forallλ>0. Since gis nilpotent, G=expP(g). Let x∈G∩¯ Q;thenx=expP(v) for some v∈gand limλ→∞ δλ(x)= expP(limλ→∞ δλv) =eP. It follows that the curve γ:(0,1]→ ¯ P,γ(t)=δt−1x, extends to a continuous path [0,1]→ ¯ Pconnecting γ(0)=ePto γ(1)=xand laying in G.Since δλ(x)∈¯ Qfor all λ>0, then γlies in ¯ Qas well. Since g⊕q=p, there are open neighborhoods U⊂gand V⊂qof 0 such that =expP(U)expP(V)is an open neighborhood of ePin ¯ Pand the following holds: The connected component of ∩Gcontaining ePis expP(U), the connected component of ∩¯ Qcontaining ePis expP(V),andexp P(U)∩expP(V)={eP}. Since γjoins xto ePcontinuously, then γ([0,1])∩lies in both the connected components of ∩Gand ∩¯ Qcontaining eP, i.e., γ([0,1])∩⊂expP(U)∩expP(V)={eP}. This implies that x=eP. Lemma 2.3 (Lemma on Lie groups) Let G be a Lie subgroup of a Lie group P and let ι:G→P the inclusion. The image ι(G)is not closed in P if and only if there is a sequence 123
56 Page 6 of 22 Geometriae Dedicata (2024) 218 :56 {gn}n∈N⊂G such that limn→∞ gn=∞(i.e., gneventually escapes every compact set of G) and limn→∞ ι(gn)=eP. Proof Recall that Gis closed in Pif and only if ιis an embedding. So, if such a sequence exists then ι(G)is not closed in P. We need to prove the converse implication. Let ρbe any left-invariant Riemannian distance on G.Thenρis complete and in particular closed balls are compact. Let {gn}n∈N⊂Gbe a sequence such that limn→∞ ι(gn)=p∈P. If there is R>0 such that ρ(eG,gn)≤Rfor all n, then there is a subsequence gnkconverging to some g∞∈G. Since the immersion ι:G→Pis continuous, we obtain ι(g∞)=p, hence p∈ι(G). So, if ι(G)is not closed, then there is a sequence {gn}n∈N⊂Gsuch that limn→∞ ι(gn)= p∈Pbut gn→∞in G.Let{gnk}kbe a subsequence such that ρ(gnk,gnk+1)>kfor k∈N and define hk=g−1 nkgnk+1.Thenhk→∞in G, because ρ(eG,hk)=ρ(eG,g−1 nkgnk+1)= ρ(gnk,gnk+1)>kfor all k.However,ι(hk)=ι(g−1 nk)ι(gnk+1)→p−1pin Pas k→∞. Lemma 2.4 The immersed group G is closed in ¯ P. Proof We prove that, if {vn}n∈N⊂gis a sequence so that limn→∞ expP(vn)=eP,then limn→∞ vn=0. By Lemma 2.3 and expP(g)=G, this claim implies that Gis closed in P. Let {vn}n∈N⊂gbe a sequence with limn→∞ expP(vn)=eP.LetU⊂gand W⊂qbe open neighborhoods of 0 such that the map U×W→P,(u,w) → expP(u)expP(w) is a diffeomorphism onto its image. Then, for nlarge enough, there are un∈Uand wn∈W so that expP(un)expP(wn)=expP(vn). Therefore, expP(un)−1expP(vn)=expP(wn)∈ ¯ Q∩G. By Lemma 2.2,wehaveexp P(un)=expP(vn). By Lemma 2.1,wehaveun=vn. Since expP(un)→eP,thenvn=un→0. Corollary 2.5 The immersed group ¯ Qisclosedin ¯ P. Proof This is a consequence of Lemma 2.4 and part (iii) of Lemma 2.15 in [4] Since ¯ Qis closed, we may consider the homogeneous manifold M:= ¯ P/¯ Qwith quotient projection π:¯ P→M. The action of ¯ Pmay have a non-trivial kernel K:= {p∈¯ P:p.x=x∀x∈M}= p∈¯ P p¯ Qp −1. Lemma 2.6 The kernel K of the action of ¯ P on M is discrete and contained in ¯ Q. Moreover, if p ∈K,thenδλp=p for all λ>0. Proof Clearly Kis a normal and closed subgroup of ¯ Pand it is contained in ¯ Q.Letv∈ Lie(K),theLiealgebraofK.Then forsomepositiveinteger, wemaywritev=v0+···+v, with vi∈gifor every i=0,...,.SinceLie(K)is an ideal in pcontained in q, it follows in particular that for all i=0,...,, [[...[[vi,y1],y2],...],y+1]∈gi−−1∩q={0}, for every y1,...,y+1∈g−1. By definition of Tanaka prolongation, this implies that v=0. Therefore, the Lie algebra of Kis trivial and so Kis discrete. Since K=x∈¯ Px¯ Qx−1, it is clear that δλ(K)⊂Kfor all λ>0. However, since λ→ δλpis a continuous curve passing through p,wemusthaveδλp=pwhen p∈K. 123
Geometriae Dedicata (2024) 218 :56 Page 7 of 22 56 From Lemma 2.6 it follows that P:= ¯ P/Kand Q:= ¯ Q/Kare Lie groups, that Qis closed in Pand M=P/Q. Moreover, the maps δλare automorphisms of Pas well, for all λ>0. Since G∩K={e}, the group Gis embedded in Pwith G∩Q={e}. Remark 2.7 If we are given Gand Qinside P, for instance as matrix groups, we may want to visualise the action of Pon Mas a local action of Pon G. In other words, if p∈P, then there may be open subsets Up,Vp⊂Gand a distribution-preserving diffeomorphism fp:Up→Vpthat corresponds to the action of pon M, i.e., fp(g1)is the only g2∈G, if it exists, such that {g1Q)∩G={g2}. In general, such construction is not possible for all p∈P,butifpis near enough to eP,thenUp,Vpand fpdo exist. The fact that such fpis a distribution-preserving diffeomorphism will be proved in Proposition 2.8. 2.3 Polarizations on G,Pand M We denote by π:P→Mthe quotient map, with M=P/Q.Ifp∈Pand m∈M,we use the notation p.mor p(m)for the action of pon m. In such contexts, we will identify elements p∈Pwith smooth diffeomorphisms p:M→M. Recall that on Gwe have the polarisation Gwith (G)e=g−1. We define on Pthe polarisation Psuch that (P)eP=g−1⊕q. Notice that G=P∩TG.DefineM:= dπ(P)which is a subset of TM. We shall prove that Mis a P-invariant polarisation on M. Proposition 2.8 The set M⊂T M is a P-invariant, bracket generating subbundle of M. In particular, (M,M)is a polarised manifold and the diffeomorphisms p :M→Mfor p∈P are distribution-preserving diffeomorphisms. Proof Noticethat Mis a P-invariant subsetof TM.In orderto show thatMis asubbundle, we need to prove that, if p1,p2∈Pare such that π(p1)=π(p2),then dπ((P)p1)=dπ((P)p2). (1) Since p◦π=π◦Lpfor all p∈P,then(1) is equivalent to d(p−1 2◦π◦Lp1)[(P)e]= dπ[(P)e].Letp=p1and choose q∈Qso that p2=p1q.Thenp−1 2◦π◦Lp1=π◦Lq−1 and thus (1) is also equivalent to Adq[(P)e]mod q=(P)emod q.(2) Since Ad is a homomorphism and every q∈Qis the finite product of exponential elements, it’s enough that we show (2)forq=exp y,y∈q. Denote by y0the projection of yon g0. Let w∈g−1⊕qand denote by w−1its projection on g−1.Then Adqwmod q=ead(y)wmod q =ead(y0)w−1mod q. Since ead(y0):g−1→g−1is a bijection, we conclude that Adq[(P)e]mod q=g−1 mod q. This proves (2) and therefore (1). Finally, we need to show that Mis bracket generating. Recall that, for an analytic subbundle of an analytic manifold, being bracket generating is equivalent to being connected by curves tangent to the subbundle, and that quotients of Lie groups and invariant subbundles are all analytic. Thus, let m0=π(p0)and m1=π(p1)in M.ThenthereisaC1-curve γ:[0,1]→Psuch that γ(0)=p0,γ(1)=p1and γ(t)∈Pfor all t∈[0,1]. Hence, the curve π◦γ:[0,1]→Mgoes from m0to m1and is clearly tangent to M. 123
56 Page 8 of 22 Geometriae Dedicata (2024) 218 :56 Proposition 2.9 The restriction π|G:(G, G)→(M,M)is a distribution-preserving diffeomorphism onto its image, which is an open subset of M. Proof First, we show that π|Gis injective. Let a,b∈Gsuch that π(a)=π(b).Then π(e)=π(a−1a)=a−1.π(a)=a−1π(b)=π(a−1b), i.e., a−1b∈Q.Sincea−1b∈Gand G∩Q={e},thena=b. Second, we show that π|Gis an immersion. Since ker(dπ|g)=dLg(q)and dLg(q)∩ TgG=DLg(q)∩DLg(g)={0},thend(π|G)|g=(dπ|g)|TgGis injective, for all g∈G. Third,weclaimdπ|G(G)=M∩T(π(G)).SinceG⊂PandsinceM=dπ(P) by definition, it follows that dπ|G(G)⊂M∩T(π(G)). Moreover, since Mis Pinvariant by Proposition 2.8,forallx∈M,dim(M)x=dim(M)π(e)=dim((g−1⊕ q)/q)=dim(g−1). Therefore, we obtain the claim by comparing the dimensions. Finally, the fact that π(G)is open in Mand the fact that π|Gis an embedding are both consequences of (π|G)being an immersion and the fact that Mand Ghave the same dimension. Remark 2.10 A first consequence of Proposition 2.9 is that any local distribution-preserving diffeomorphism on Mis in fact a local distribution-preserving diffeomorphism on G. Indeed, by the action of Pon Mand via the map π|G, any local distribution-preserving diffeomorphism of Mdefines a local distribution-preserving diffeomorphism of G. Similarly, contact vector fields on Mdefine contact vector fields on G. In case Gis a rigid stratified group and pis the full Tanaka prolongation of g,these relations are stronger, see Sect.3. 3 Distribution-preserving diffeomorphisms of Mwhen Gis rigid This section contains Theorem 3.3 for distribution-preserving diffeomorphisms in the rigid case. Relative to a vector X∈TeP,wedenoteby ˜ Xthe left-invariant vector field ˜ X(p)= dLp|e[X], and by X†the right-invariant vector field X†(p)=dRp|e[X]. Similarly, we denote by ˜ pthe Lie algebra of left-invariant vector fields and by p†the Lie algebra of rightinvariant vector fields on P. Moreover, as in the previous sections, the manifold Mis the quotient P/Qand we denote by othe point π(e)∈M. Lemma 3.1 Let :p→pbe a Lie algebra automorphism with (q)=qand (g−1⊕q)= g−1⊕q. Then there is a unique distribution-preserving Lie group automorphism L :P→P with L∗=and a unique distribution-preserving diffeomorphism Lπ:M→M with Lπ◦π=π◦L. Proof If :p→pis a Lie algebra automorphism with (q)=q, then the induced Lie group automorphism ¯ L:¯ P→¯ Phas the property that ¯ L(K)=K,whereKis the kernel of the action of ¯ Pon M. It follows that there is a Lie group automorphism L:P→Psuch that L∗=. If L:P→Pis a Lie group automorphism with L(Q)=Q, then it is well known that there is a unique diffeomorphism Lπ:M→Msuch that Lπ◦π=π◦L. Now, suppose that (g−1⊕q)=g−1⊕q, i.e., L∗(P)e=(P)e.SincePis leftinvariant, then Lis a distribution-preserving diffeomorphism of (P,P).Finally,weprove that Lπis a distribution-preserving diffeomorphism. Let X∈Pand x∈P.Then dLπ|π(x)[dπ|x[˜ Xx]] = d(Lπ◦π)|x[˜ Xx]=d(π ◦L)|x[˜ Xx]∈M|Lπ(x). 123
Geometriae Dedicata (2024) 218 :56 Page 15 of 22 56 First, case (A.1) is not isomorphic to the others because in case (A.1) we have s(2)= span{f3}while in all other three cases we have s(2)=span{f2,f3}. Second,case(A.4) is notisomorphic to theothers because incase (A.4) wehave [, s−1] ⊂ s−1while in all other cases we have [, s−1]⊂s−1. Third, for different choices of α∈Rin case (A.4) we get non-isomorphic polarised Lie algebras: To prove this, we shall show that the parameter αis independent of the choice of the basis. So, suppose that g1,g2,g3∈sform another basis with s−1=span{g1,g2}, [g1,g2]=g3,[g2,g3]=0and[g1,g3]=αg2+g3. Then one easily shows that g1= xf 1+yf2,g2=μf2and g3=λf3,forsomex,y,λ,μ ∈Rwith xμ λ=1. Moreover, [g1,g3]=αxμ λg2+xg3, which implies x=1andα=α. Finally, cases (A.2) and (A.3) are not isomorphic to each other, because in case (A.2) it holds ad f1|2 s(2)=Id|s(2), while while in case (A.3) it holds ad f1|2 s(2)=−Id|s(2). Proof of Theorem 5.1 Let us fix the notation for the Heisenberg Lie algebra. Fix a basis e1,e2,e3so that [e1,e2]=e3, and choose g−1=span{e1,e2}. The space Der(g)of the strata preserving derivations of gmay be identified with gl(2,R). First, we consider g0:= {D∈Der(g):D(e1)⊆Re1and D(e2)⊆Re2}. In this case, Prol(g,g0)=sl(3,R)=g⊕q(see, e.g., [3]), where gis identified with the Lie algebra generated by e1=⎛ ⎝ 010 000 000 ⎞ ⎠,e2=⎛ ⎝ 000 001 000 ⎞ ⎠,e3=⎛ ⎝ 001 000 000 ⎞ ⎠.(6) and qis the set of matrices in sl(3,R)of the form ⎛ ⎝ ∗00 ∗∗0 ∗∗∗ ⎞ ⎠ The modifications of gin sl(3,R)are the subalgebras of sl(3,R)of the form {X+σ(X): X∈g}, for some linear map σ:g→q. We show that all three dimensional Lie algebras with a bracket generating plane are graphs of such a σ: Case (A): If sis solvable, then define σby the assignments: σ(e1)=⎛ ⎜ ⎝ 2β 300 α−β 30 00−β 3 ⎞ ⎟ ⎠,σ(e2)=σ(e3)=0. It is easy to check that vectors fi:= ei+σ(ei),i=1,2,3, satisfy the bracket relations of case (A) in Proposition 5.2. Case (B): For this case, we choose σ(e1)=⎛ ⎝ 000 −100 000 ⎞ ⎠,σ(e2)=⎛ ⎝ 000 000 0−10 ⎞ ⎠,σ(e3)=⎛ ⎝ 000 000 −100 ⎞ ⎠. 123
56 Page 16 of 22 Geometriae Dedicata (2024) 218 :56 Case (C): we obtain the brackets in (C) by choosing σ(e1)=0,σ(e2)=⎛ ⎝ 000 1/200 000 ⎞ ⎠,σ(e3)=⎛ ⎝ 1/200 0−1/20 000 ⎞ ⎠. Case (D): In this case we use the finite prolongation su(2,1)of the Heisenberg algebra, as in [8, p313]. Let J=⎛ ⎝ 10 0 0−10 00−1⎞ ⎠. The Lie algebra su(2,1)isgivenby3×3 complex matrices Awith zero trace and such that A∗J+JA =0, where A∗is the hermitian transpose of A. Define the Lie algebra automorphism θ:su(2,1)→su(2,1),θA:= JAJ.Define X=⎛ ⎝ 0i0 −i0−i 0−i0⎞ ⎠Y=⎛ ⎝ 010 101 0−10 ⎞ ⎠Z=⎛ ⎝ 2i02i 000 −2i0−2i⎞ ⎠ H=⎛ ⎝ 001 000 100 ⎞ ⎠U=⎛ ⎝ i00 0−2i0 00i⎞ ⎠ θX=⎛ ⎝ 0−i0 i0−i 0−i0⎞ ⎠θY=⎛ ⎝ 0−10 −101 0−10 ⎞ ⎠θZ=⎛ ⎝ 2i0−2i 00 0 2i0−2i⎞ ⎠ The grading of su(2,1)is g−2(g)=span{Z} g−1(g)=span{X,Y} g0(g)=span{H,U} g1(g)=span{θX,θY} g2(g)=span{θZ}, where g−2(g)⊕g−2(g)=gis the Heisenberg Lie algebra: notice that [X,Y]=Zwhile [X,Z]=[Y,Z]=0. So, q=span{H,U,θX,θY,θZ}.Defineσ:g→qby setting σX:= − 1 16θX+i9 16θY=⎛ ⎝ 0−i1 20 −i5 80i5 8 0−i1 20⎞ ⎠, σY:= −i9 16θX−1 16θY=⎛ ⎝ 0−1 20 5 80−5 8 0−1 20⎞ ⎠, σZ:= −i9 4H+1 4U−5 16θZ=⎛ ⎝ −i3 80−i13 8 0−i1 20 −i23 80i7 8 ⎞ ⎠. One can easily check that f1=X+σX,f2=Y+σYand f3=Z+σZform a basis of a Lie subalgebra of su(2,1)satisfying the relations of Case (D). 123
Geometriae Dedicata (2024) 218 :56 Page 17 of 22 56 Remark 5.4 The map σabove can easily be found using the software Maple and it is not unique. 5.1.1 Rigid motions of the plane as a modification of the Heisenberg group We conclude this section discussing more in detail the case of the group of rigid motions of the plane as a modification of the Heisenberg group. At a group level, we may represent points in the Heisenberg group Has matrices in SL(3,R)by H(x1,x2,x3):= ⎛ ⎝ 1x1x3 01x2 00 1 ⎞ ⎠, for x1,x2,x3∈R. The Lie algebra of the the group of rigid motions of the plane E(2)corresponds to the case (A) with α=−1andβ=0. The corresponding representation in sl(3,R)giveninthe previous theorem is the span of the vectors f1=⎛ ⎝ 010 −100 000 ⎞ ⎠,f2=⎛ ⎝ 000 001 000 ⎞ ⎠,f3=⎛ ⎝ 001 000 000 ⎞ ⎠. At the group level, the points of E(2)inside SL(3,R)are parametrized by R(y1,y2,y3):= ⎛ ⎝ cos y1sin y1y3 −sin y1cos y1y2 001 ⎞ ⎠ where y1∈R/(2πZ)and y2,y3∈R. With the procedure described in Remark 4.3, we find the mapping E(2)→H: R(y1,y2,y3)→ H(tan y1,y2,y3), which is defined on the domain (−π/2,π/2)×R2. 5.2 Modifications of the free nilpotent Lie group F24 We consider the free nilpotent Lie algebra f24 =span{ei:i=1,...,8}of rank 2 and step 4 and the corresponding simply connected Lie group F2,4. We will prove the following result Theorem 5.5 There exists a nilpotent Lie group S, not isomorphic to F2,4, that is a modification of F2,4and is globally equivalent to F2,4. Proof The Lie brackets in f24 are [e2,e1]=e3,[e3,e1]=e4,[e3,e2]=e5, [e4,e1]=e6,[e5,e1]=e7,[e4,e2]=e7,[e5,e2]=e8. ItisknownthatthefullTanakaprolongationoff24 isp=f24⊕Der(g),withDer(g)≃gl(2,R) (see [17]). Therefore, the modifications of f24 are subalgebras of pthat are graphs of some 123
56 Page 18 of 22 Geometriae Dedicata (2024) 218 :56 linear map σ:f24 →gl(2,R). Here we only consider σthat on the basis of f24 is zero except for σ(e1). Imposing that the graph is a Lie algebra, a direct computation shows that σ(e1)=a0 cb , where a,b,c∈R. We obtain a three parameter family s(a,b,c)of Lie algebras with basis f1,..., f8,where f1=e1+σ(e1)and fi=eifor i=2,...,8, and brackets [f2,f1]=f3−bf2,[f3,f1]=f4−(a+b)f3,[f3,f2]=f5, [f4,f1]=f6−cf5−(2a+b)f4,[f4,f2]=f7,[f5,f1]=f7−(a+2b)f5, [f1,f6]=2cf7+(3a+b)f6,[f1,f7]=cf8+2(a+b)f7, [f1,f8]=(a+3b)f8,[f5,f2]=f8. In particular, setting a=b=0 gives a one parameter family of nilpotent Lie algebras s(c). We now find the distribution-preserving diffeomorphism from S(c)to F24 when c=1, as in Remark 4.3. Every point in S(1)is of the form expP(xifi). Following [12], expP(xifi)=(EF24 (x1σ(e1);xiei), expGL(x1σ(e1)) ∈F24 GL(2,R),where EF24 (x1e1;xiei)=γ(1)and γ:[0,1]→F24 is the solution of γ(t)=dLγ(t)expGL(tx1σ(e1))(xiei) γ(0)=eF24 . The image of this point via is going to be that element p∈F24 such that gQ = expP(xifi)Q, i.e., expPxifi=EF24 (x1e1;xiei). To compute this, we first observe that v:= expGL(tx1σ(e1)) xiei =x1,x2 1t+x2,x3,x4,x5+tx1x4,x6,x7+2tx1x6,x8+tx1x7+t2x2 1x6. Second, we need to compute dLγvusing the Baker–Campbell–Hausdorff formula: dLγv=d dhh=0 exp−1(exp(γ ) exp(hv)) =v+1 2[γ,v]+ 1 12[γ,[γ,v]]. 123
Geometriae Dedicata (2024) 218 :56 Page 19 of 22 56 The system of differential equations ˙γ=dLγvthat we obtain is ˙γ1=x1 ˙γ2=tx2 1+x2 ˙γ3=−1 2tx2 1γ1−1 2x2γ1+1 2x1γ2+x3 ˙γ4=1 12tx2 1γ2 1+1 12x2γ2 1−1 12(γ1γ2−6γ3)x1−1 2x3γ1+x4 ˙γ5=1 12x2γ1γ2−1 12x1γ2 2+1 12x2 1γ1γ2+6x2 1γ3+12x1x4t−1 2x3γ2 +1 2x2γ3+x5 ˙γ6=1 12x3γ2 1−1 12(γ1γ3−6γ4)x1−1 2x4γ1+x6 ˙γ7=1 6x3γ1γ2−1 12x2γ1γ3−1 12 x2 1γ1γ3+6x1x4γ1−6x2 1γ4−24x1x6t −1 12 (γ2γ3−6γ5)x1−1 2x5γ1−1 2x4γ2+1 2x2γ4+x7 ˙γ8=t2x2 1x6+1 12x3γ2 2−1 12x2γ2γ3−1 12 x2 1γ2γ3+6x1x4γ2−6x2 1γ5−12x1x7t −1 2x5γ2+1 2x2γ5+x8. Third, we need to integrate this system of ODEs with initial conditions γi(0)=0for every i=1,...,8. The solution is γ1(t)=tx1 γ2(t)=1 2t2x2 1+tx2 γ3(t)=−1 12t3x3 1+tx3 γ4(t)=tx4 γ5(t)=− 1 240t5x5 1+1 12t3x2 1x3+1 2t2x1x4+tx5 γ6(t)=1 720t5x5 1+tx6 γ7(t)=1 720t6x6 1+1 360t5x4 1x2+t2x1x6+tx7 γ8(t)=1 5040t7x7 1+1 720t6x5 1x2+1 720 x3 1x2 2+3x4 1x3t5 −1 12 x1x2x4−x2 1x5−4x2 1x6t3+1 2t2x1x7+tx8. Therefore, the mapping from S(1)to Gis :expP(xifi)→ γ(1), which is a global, surjective smooth distribution-preserving diffeomorphism. Finally, S(1)is not isomorphic to F2,4because S(1)has nilpotency step 5 instead of 4, as one can easily see from the expression of the Lie brackets in s(1). 123
56 Page 20 of 22 Geometriae Dedicata (2024) 218 :56 Remark 5.6 The mapping from S(1)to Gdescribed above is in particular bi-Lipschitz on every compact set, when the groups are endowed with left-invariant sub-Riemannian distances. Notice, however, that this is not a global quasiconformal mapping. 5.3 Modifications of ultra-rigid stratified groups A stratified Lie algebra gis called ultra-rigid if the only automorphisms of gpreserving the stratifications are dilations, see [10]. In particular, the full Tanaka prolongation of such gis p=gR, as semi-directproductof Liealgebras. Inthis section wedescribe all modifications in gRand their equivalence relation. Many results do not need the assumption of gbeing ultra-rigid, so we assume this hypothesis only when needed. Let g=−1 j=−sgjbe a stratified Lie algebra. Let D:g→gbe the linear map with Dv=jvfor v∈g−j. Notice that Dis a derivation of gthat preserves the layers and that δet=etD :g→gare the dilations. The semi-direct product p:= gRis the Lie algebra whose Lie brackets are [(0,a), (Y,0)]=(aDY,0)hence [(X,a), (Y,b)]=([X,Y]+aDY −bDX,0). Proposition 5.7 Let σ:g→Rbe a linear map and set s:= {(X,σX):X∈g}. The vector space sis a Lie subalgebra of gRif and only if −s j=−2gj⊂ker σ. Proof First, we note that sis a Lie algebra if and only if, for all X,Y∈g, σ([X,Y])+(σ X)(σ DY)−(σY)(σ DX)=0.(7) ⇒Suppose sis a Lie algebra, i.e., (7) holds for all X,Y∈g. We prove −s j=−2gj⊂ ker σby induction on j.IfX,Y∈g−1,thenDX =Xand DY =Y, thus (7) implies σ([X,Y])=0. Since g−2=[g−1,g−1], it follows that g−2⊂ker σ. Now, suppose that g−k⊂ker σfor k≥2. If X∈g−1and Y∈g−k,then(7) implies that σ([X,Y])=0. Since g−k−1=[g−1,g−k], it follows that g−k−1⊂ker σ. We conclude that −s j=−2gj⊂ker σ. ⇐Suppose −s j=−2gj⊂ker σ. By the bilinearity of the expression, we need to show that (7) holds only when X∈giand Y∈gjfor some iand j.Sinceσis non-zero only on the first layer, the only non-trivial instance of (7)isforX,Y∈g−1. In this case, σ([X,Y])=0, and (σ X)(σ DY)−(σY)(σ DX)=(σ X)(σY)−(σ Y)(σ X)=0. Therefore, (7) is satisfied and sis a Lie algebra. Lemma 5.8 The Lie algebra automorphisms φ:p→psuch that φ({0}×R)={0}×R and φ(g−1×R)=g−1×Rare exactly those of the form φ(X,a)=(φ1X,a)for some Lie algebra automorphism φ1:g→gthat preserves the layers. Proof On the one hand, if φ1:g→gis a Lie algebra automorphism that preserves the layers, then φ(X,a)=(φ1X,a)is clearly a Lie algebra automorphism φ:p→pwith φ({0}×R)={0}×Rand φ(g−1×R)=g−1×R, because φ1D=Dφ1. On the other hand, if φ:p→pis a Lie algebra automorphism, then φ(g×{0})=g×{0} because g×{0}=[p,p]. Suppose also that φ({0}×R)={0}×Rand φ(g−1×R)=g−1×R. Then φ(X,a)=φ(X,0)+φ(0,a)=(φ1(X), 0)+(0,φ 2(a)) and φ1(g−1)=g−1.This implies that φ1(gj)=gjfor all j, as one can prove by induction on j. Notice that, for all X∈gand all a∈R, φ2(a)Dφ1X=[(0,φ 2(a)), (φ1X,0)]=φ([(0,a), (X,0)])=φ(aDX,0)=aφ1DX. For every X∈g−1,DX =Xand Dφ1X=φ1X, hence φ2(a)φ1X=aφ1X, i.e., φ2(a)=a. 123
Geometriae Dedicata (2024) 218 :56 Page 21 of 22 56 Theorem 5.9 Suppose that gis ultra-rigid, i.e., p=gRis its full Tanaka prolongation. The set of all non-isomorphic modifications of gis parametrized by g∗ −1/R>0. Moreover, all modifications of gin pare solvable and the only nilpotent one is gitself. Proof The set of all modifications of gin pcan be identified with g∗ −1by Proposition 5.7, where σ∈g∗ −1is identified with σ(jvj)=σ(v−1)for jvj∈gand the modification sσ:= {(X,σX):X∈g}⊂p.Sincegis rigid, by Theorem 4.6 two modifications σ, τ ∈ g∗ −1are isomorphic if and only if there is a Lie algebra automorphism φ:p→pwith φ({0}×R)={0}×Rand φ(g−1×R)=g−1×Rsuch that φ(sσ)=sτ. Therefore, by Lemma 5.8, two modifications σ, τ ∈g∗ −1are isomorphic if and only if there is a Lie algebra automorphism φ1:g→gsuch that, for all X∈g, (φ1X,σX)=(φ1X,τφ 1X), i.e., σX=τφ1Xfor all X∈g−1.Now,sincegis ultrarigid, φ1=δλfor some λ>0. Therefore, two modifications σ, τ ∈g∗ −1are isomorphic if and only if there is λ>0such that σ=λτ. Finally, notice that all modifications of gin pare solvable, because pitself is solvable. Moreover, the only nilpotent modification is gitself. Indeed, if s= g, then there is X∈g−1 with σX= 0, so that, if Y∈g−sis nonzero, then [(X,σX), (Y,0)]=sσX(Y,0),where sis the step of g. Therefore, we obtain that (Y,0)∈[s,[...,[s,s]...]] for any order of brackets, that is, sis not nilpotent. Author Contributions SebastianoNicolussi GoloandAlessandroOttazziwereequallyinvolvedintheresearch that led to this manuscript and in writing it. Funding Open Access funding enabled and organized by CAUL and its Member Institutions. Declarations Conflict of interest The authors declare no competing interests. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Agrachev, A., Barilari, D.: Sub-Riemannian structures on 3D Lie groups. J. Dyn. Control Syst. 18(1), 21–44 (2012) 2. Baudoin, F., Cecil, M.: The subelliptic heat kernel on the three-dimensional solvable Lie groups. Forum Math. 27(4), 2051–2086 (2015) 3. Cowling, M., et al.: Contact and conformal maps on Iwasawa Ngroups. In: Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 13.3-4 (2002). Harmonic analysis on complex homogeneous domains and Lie groups (Rome, 2001), pp. 219–232 4. Cowling, M.G., et al.: From homogeneous metric spaces to Lie groups. arXiv:1705.09648 e-prints (2017) 5. Gordon,C.S.,Wilson,E.N.:IsometrygroupsofRiemanniansolvmanifolds.Trans.Am.Math.Soc.307(1), 245–269 (1988) 123
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