Homogeneous hypersurfaces and totally geodesic submanifolds

INTERNATIONAL DOCTORAL
SCHOOL OF THE USC
Albe o
Rod íguez Vázquez
PhD Thesis
Homogeneous hype su aces
and o ally geodesic
submani olds
San iago de Compos ela, 2022
Doc o al P og amme in Ma hema ics
TESE DE DOUTORAMENTO
HOMOGENEOUS
HYPERSURFACES AND
TOTALLY GEODESIC
SUBMANIFOLDS
Albe o Rod ´ıguez V´azquez
ESCOLA DE DOUTORAMENTO INTERNACIONAL
PROGRAMA DE DOUTORAMENTO EN MATEM´
ATICAS
SANTIAGO DE COMPOSTELA
ANO 2022
.
DECLARACI ´
ON DO AUTOR DA TESE
Homogeneous hype su aces
and o ally geodesic submani olds
D. Albe o Rod ´ıguez V´azquez
P esen o a mi˜na ese, seguindo o p ocedemen o adecuado ao Regulamen o, e
decla o que:
1) A ese aba ca os esul ados da elabo aci´on do meu aballo.
2) No seu caso, na ese aise e e encia ´as colabo aci´ons que i o es e
aballo.
3) A ese ´e a e si´on de ini i a p esen ada pa a a s´ua de ensa e coincide
coa e si´on en iada en o ma o elec ´onico.
4) Con i mo que a ese non inco e en ning´un ipo de plaxio dou os au-
o es nin de aballos p esen ados po min pa a a ob enci´on dou os
´ı ulos.
En San iago de Compos ela, 1 de xullo de 2022.
Asdo. Albe o Rod ´ıguez V´azquez

.
AUTORIZACI ´
ON DO DIRECTOR DA TESE
Homogeneous hype su aces
and o ally geodesic submani olds
D. Jos´e Ca los D´ıaz Ramos
INFORMA:
Que a p esen e ese se co esponde co aballo ealizado po D. Albe o Rod ´ıguez
V´azquez, baixo a mi˜na di ecci´on, e au o izo a s´ua p esen aci´on, conside ando que
e´une os equisi os esixidos no Regulamen o de Es udos de Dou o amen o da USC,
e que como di ec o des a non inco o nas causas de abs enci´on es ablecidas na lei
40/2015.
De aco do co indicado no Regulamen o de Es udos de Dou o amen o, decla a
am´en que a p esen e ese de dou o amen o ´e id´onea pa a se de endida en base ´a
modalidade de Monog ´a ica con ep oducci´on de publicaci´ons, nos que a pa icipaci´on
do dou o ando oi decisi a pa a a s´ua elabo aci´on e as publicaci´ons se axus an ao Plan
de In es igaci´on.
En San iago de Compos ela, 1 de xullo de 2022.
Asdo. Jos´e Ca los D´ıaz Ramos
.
.
AUTORIZACI ´
ON DO DIRECTOR DA TESE
Homogeneous hype su aces
and o ally geodesic submani olds
D. Miguel Dom´ınguez V´azquez
INFORMA:
Que a p esen e ese se co esponde co aballo ealizado po D. Albe o Rod ´ıguez
V´azquez, baixo a mi˜na di ecci´on, e au o izo a s´ua p esen aci´on, conside ando que
e´une os equisi os esixidos no Regulamen o de Es udos de Dou o amen o da USC,
e que como di ec o des a non inco o nas causas de abs enci´on es ablecidas na lei
40/2015.
De aco do co indicado no Regulamen o de Es udos de Dou o amen o, decla a
am´en que a p esen e ese de dou o amen o ´e id´onea pa a se de endida en base ´a
modalidade de Monog ´a ica con ep oducci´on de publicaci´ons, nos que a pa icipaci´on
do dou o ando oi decisi a pa a a s´ua elabo aci´on e as publicaci´ons se axus an ao Plan
de In es igaci´on.
En San iago de Compos ela, 1 de xullo de 2022.
Asdo. Miguel Dom´ınguez V´azquez

Abs ac
This Ph.D. hesis deals wi h he s udy o ce ain classes o submani olds in he p es-
ence o symme y. Namely, esul s ha e been de i ed ega ding he heo y o sub-
mani olds in Riemannian homogeneous spaces wi h a special emphasis on symme ic
spaces. In his disse a ion, we will ocus on wo o he mos na u al classes o sub-
mani olds ha one can s udy in Riemannian mani olds. These a e homogeneous
hype su aces and o ally geodesic submani olds.
Rega ding he i s ones, we will conclude he classi ica ion o homogeneous hype -
su aces in symme ic spaces o ank one, by inishing he classi ica ion in qua e nionic
hype bolic spaces. As o o ally geodesic submani olds, we will de i e di e en classi-
ica ions. In pa icula , we will classi y o ally geodesic submani olds in he ollowing
spaces: in p oduc s o symme ic spaces o ank one, in excep ional symme ic spaces,
and in Hop -Be ge sphe es.
x
Objec i es and hypo heses
This hesis has he ollowing hypo heses and objec i es:
H1: In he con ex o spaces wi h cons an sec ional cu a u e he concep s o
isopa ame ic hype su ace and hype su ace wi h cons an p incipal cu a u es a e
equi alen . Howe e , in mo e gene al ambien spaces, he e a e examples o isopa a-
me ic hype su aces which do no ha e cons an p incipal cu a u es.
O1: Cons uc examples o non-isopa ame ic hype su aces wi h cons an p in-
cipal cu a u es.
H2: The p oblem o classi ying cohomogenei y one ac ions on qua e nionic hy-
pe bolic spaces has been open o almos wen y yea s. A solu ion o i would yield
he classi ica ion o hese ac ions on symme ic spaces o ank one. In 2001, Be nd
and B ¨uck ound a me hod o cons uc cohomogenei y one ac ions on non-compac
symme ic spaces o ank one. La e , Be nd and Tama u classi ied hese ac ions on
he complex hype bolic space and on he Cayley hype bolic plane.
O2: Classi y cohomogenei y one ac ions on qua e nionic hype bolic spaces o
conclude he classi ica ion o cohomogenei y one ac ions on symme ic spaces o ank
one.
H3: The classi ica ion o o ally geodesic submani olds in symme ic spaces o
ank one is a classical and well-known esul . I is clea ha in a p oduc o sym-
me ic spaces o ank one, he ex insic p oduc o o ally geodesic submani olds is
again o ally geodesic. Howe e , apa om hese ob ious examples he e could be
many mo e o ally geodesic submani olds and we lack an e ec i e way o unde s and
he moduli space o o ally geodesic submani olds in a bi a y p oduc s o ank one
symme ic spaces.
O3: Classi y o ally geodesic submani olds in a bi a y p oduc s o ank one sym-
me ic spaces and de elop an e ec i e way o unde s and he moduli space o such
o ally geodesic submani olds.
H4: In He mi ian symme ic spaces o ank one, ha is, complex hype bolic and
p ojec i e spaces, e e y o ally geodesic submani old has cons an K¨ahle angle equal
o 0 o π/2. Klein ound examples o o ally geodesic submani olds in He mi ian
symme ic spaces o ank wo wi h non- i ial cons an K¨ahle angle (i.e. di e en
om 0 and π/2).
x ii
x iii Objec i es and hypo heses
O4: Cons uc new examples o o ally geodesic submani olds wi h non- i ial
cons an K¨ahle angle.
H5: The ou s anding p oblem o classi ying o ally geodesic submani olds in i e-
ducible symme ic spaces has only been sol ed o ank one and ank wo.
O5: De elop new ools o ackle he a o emen ioned p oblem and o ob ain new
classi ica ion esul s when he ank is g ea e han wo.
H6: The s udy o o ally geodesic submani olds in homogeneous spaces is ha de
han in he case o symme ic spaces o a ious easons. We lack classi ica ion esul s.
O6: De elop new ools o s udy o ally geodesic submani olds in homogeneous
spaces and s a a p og am o classi y o ally geodesic submani olds in special amilies
o homogeneous spaces such as homogeneous spaces homeomo phic o sphe es.
H7: The index o symme y is an in a ian ha measu es he ex en o which a
homogeneous space ails o be symme ic. This is known o some homogeneous spaces
bu i has no been compu ed in homogeneous spaces homeomo phic o sphe es.
O7: Compu e he index o symme y o Hop -Be ge sphe es.
Me hodology
This hesis has ollowed he common app oach o esea ch in Ma hema ics. This is:
by analyzing pa e ns, conduc ing logic analysis, and pe o ming calcula ions, we can
de i e gene al p ope ies o s uc u es.
Howe e , in o de o ca y ou hese asks, i is absolu ely necessa y o assimila e
a se ies o known ideas and concep s. Fo he aining o he au ho o his hesis he
eading and s udy o some books and a icles like [14, 32, 74, 90, 110, 150, 193] has
been undamen al.
Finally, he discussion wi h expe s in he ield o di e en ial geome y has had a
huge ele ance in his hesis as well. Apa om he discussion wi h my ad iso s, i is
impo an o poin ou he knowledge acqui ed in he ollowing esea ch s ays: h ee
mon hs a King’s College London isi ing J¨u gen Be nd , wo mon hs in S u ga
wi h And eas Koll oss, and wo mon hs in C´o doba wi h Ca los Olmos. F om all o
hese esea ch s ays many new and ele an ideas o his hesis ha e eme ged.
xix

In oduc ion
Symme y, as wide o na ow as you may de ine i s meaning, is one idea by which man
h ough he ages has ied o comp ehend and c ea e o de , beau y, and pe ec ion.
This commen is due o He mann Weyl and e eals ha symme y lies in he
e y co e o human knowledge. Pe haps, he mos na u al ield o s udy symme y is
geome y. Felix Klein desc ibed geome y as he s udy o hose p ope ies o a space
ha a e in a ian unde a ans o ma ion g oup. F om he iewpoin o Riemannian
geome y, he na u al g oup o s udy is he isome y g oup. Mo eo e , mos geome ic
objec s ha we can pe cei e by means o ou senses can be desc ibed in e ms o
cu es and su aces. Submani olds p o ide he na u al gene aliza ion o hese objec s
o highe dimensions.
This Ph.D. hesis deals wi h he s udy o ce ain classes o submani olds in he
p esence o symme y. In pa icula , we ha e de i ed esul s conce ning submani olds
in Riemannian homogeneous spaces wi h a ocus on symme ic spaces.
Roughly speaking, a homogeneous space is one ha looks he same a e e y poin .
Fo his eason, homogeneous spaces se e as model spaces o many di e en ypes
o geome ic s uc u es. Speci ically, we a e in e es ed in homogeneous spaces ha
esul om isome ic ac ions, ha is, om Lie g oup ac ions ha p ese e he me ic.
Symme ic spaces cons i u e a special class o homogeneous spaces. They occu
in a wide a ie y o si ua ions in bo h Ma hema ics and Physics. A symme ic space
is a Riemannian mani old whose g oup o isome ies con ains an in e sion symme y
a each poin . This implies ha hese spaces admi a nice desc ip ion in e ms o
Lie g oups, and ha we can use algeb aic ools o ge a deepe unde s anding o
hei geome y. Symme ic spaces we e classi ied by ´
Elie Ca an in he 1920s and
some examples a e: he Euclidean spaces, he ound sphe es, he hype bolic spaces,
G assmannians, he se o o hogonal complex s uc u es o a ec o space, he se
o inne p oduc s o a ec o space, he se o Lag angian subspaces o a symplec ic
ec o space, o compac Lie g oups.
P obably, he mos impo an in a ian in a symme ic space is he ank. The
ank is he g ea es dimension o a p ope , la , o ally geodesic submani old. Sym-
me ic spaces o ank one oge he wi h Euclidean spaces o m, up o quo ien s, a
p i ileged amily wi hin Riemannian geome y, he so-called 2-poin homogeneous
spaces, see [168]. These a e de ined as hose Riemannian mani olds Msuch ha o
e e y wo pai s o poin s (p1, p2) and (q1, q2) sa is ying d(p1, p2) = d(q1, q2), he e is
an isome y φo Msuch ha φ(pi) = qi o each i∈ {1,2}. In his hesis, symme ic
1
2 In oduc ion
spaces o ank one will play a undamen al ole.
In wha ollows, we summa ize he o iginal con ibu ions o his hesis, along wi h
he s a e-o - he-a o he ma hema ical p oblems ha mo i a ed ou in es iga ions.
A non-isopa ame ic hype su ace wi h cons an p incipal cu a u es
Two in e es ing classes o hype su aces o Riemannian mani olds a e isopa ame ic
hype su aces and hype su aces wi h cons an p incipal cu a u es. I is known [44]
ha a hype su ace in a eal space o m is isopa ame ic i and only i i has con-
s an p incipal cu a u es. This is no longe ue o o he symme ic spaces. Fo
example, he e a e hype su aces in complex hype bolic spaces ha a e isopa ame ic
bu do no ha e cons an p incipal cu a u es [60]. Howe e , i is no known i he e
exis s a hype su ace o a symme ic space wi h cons an p incipal cu a u es ha is
no isopa ame ic. This was no e en known o he gene al se ing o Riemannian
mani olds. Mo eo e , he cons uc ion o a minimal, non-isopa ame ic closed hy-
pe su ace wi h cons an p incipal cu a u es in he complex p ojec i e space would
yield a coun e example (see [83]) o he longs anding Che n conjec u e on isopa a-
me ic hype su aces in sphe es, which asse s ha a minimal closed hype su ace
wi h cons an scala cu a u e in a ound sphe e mus be isopa ame ic.
In his hesis we cons uc an explici example o a con o mally la me ic in Rn
ha admi s a o ally geodesic hype su ace (in pa icula minimal and wi h cons an
p incipal cu a u es) ha is no isopa ame ic. This p o ides he i s example o a
non-isopa ame ic hype su ace wi h cons an p incipal cu a u es in a Riemannian
mani old, and i shows ha he equi alence be ween isopa ame ici y and cons ancy
o he p incipal cu a u es in spaces o cons an cu a u e does no hold in he mo e
gene al se ing o con o mally la spaces. The main idea o his cons uc ion was
o de ine a con o mally la me ic in Rnadmi ing a o ally geodesic hype plane,
bu wi h a e y small isome y g oup ha spoils he good beha io o he pa allel
hype su aces o such hype plane.
Cohomogenei y one ac ions on ank one symme ic spaces
A cohomogenei y one ac ion on a Riemannian mani old Mis an isome ic ac ion
wi h codimension one p incipal o bi s. The p incipal o bi s o such an ac ion a e ho-
mogeneous hype su aces. The p oblem o classi ying cohomogenei y one ac ions on
a gi en space is a classical p oblem in submani old geome y ha aces back o he
imes o Beniamino Seg e [161] and ´
Elie Ca an [44], who classi ied cohomogenei y one
ac ions on Euclidean and eal hype bolic spaces, espec i ely. Much la e , Koll oss
classi ied cohomogenei y one ac ions on i educible symme ic spaces o compac ype
[113]. A e his wo k, Be nd and Tama u s a ed a p og am o s udy cohomogenei y
one ac ions on symme ic spaces o non-compac ype [26, 28, 29]. Using he ideas de-
eloped in hese a icles, Be nd and Tama u [28] we e able o classi y cohomogenei y
one ac ions on e e y symme ic space o ank one excep on qua e nionic hype bolic
spaces.
Twen y yea s a e Be nd and B ¨uck announced he i s non- i ial examples
o cohomogenei y one ac ions on qua e nionic hype bolic spaces in [13], we ha e ob-
In oduc ion 3
ained he ull classi ica ion o cohomogenei y one ac ions on qua e nionic hype bolic
spaces up o o bi equi alence. Mo eo e , as a by-p oduc o ou p oo , we ound an
uncoun able numbe o examples o inhomogeneous isopa ame ic amilies o hype -
su aces wi h cons an p incipal cu a u es. These isopa ame ic amilies cons i u e
he only such examples known in Riemannian mani olds, apa om he celeb a ed
Fe us, Ka che , and M¨unzne hype su aces in sphe es [78] and one example in he
Cayley hype bolic plane [60].
The classi ica ion o cohomogenei y one ac ions on qua e nionic hype bolic spaces
was educed o a e y in ol ed qua e nionic linea algeb a p oblem. This one boils
down o classi ying eal subspaces o a qua e nionic Euclidean ec o space Hnsuch
ha he e exis s a subg oup o Sp1Spnac ing ansi i ely on hei uni sphe es. We
call hese spaces p o ohomogeneous subspaces. In pa icula , p o ohomogeneous sub-
spaces a e in ima ely ela ed o he no ion o qua e nionic K¨ahle angle, which is a
gene aliza ion o he concep o K¨ahle angle s udied in some ecen wo ks (see o
ins ance [60]).
The main idea o sol e he p oblem men ioned abo e is o classi y subspaces wi h
cons an qua e nionic K¨ahle angle o dimension less han o equal o ou and hen
build e e y p o ohomogeneous subspace ou o hese. The key ing edien s o p o e
his a e ce ain opological and Lie g oup heo e ic ools. Mo eo e , o each p o oho-
mogeneous subspace o dimension g ea e han ou , one can can cons uc a Cli o d
s uc u e on i . The e a e wo inequi alen classes o i educible Cl3-modules, and by
mixing hem we can p oduce non-p o ohomogeneous subspaces wi h cons an qua e -
nionic K¨ahle angle whose dimension is a mul iple o ou . These induce inhomoge-
neous isopa ame ic hype su aces wi h cons an p incipal cu a u es in qua e nionic
hype bolic spaces.
To ally geodesic submani olds in p oduc s o ank one symme ic spaces
The p oblem o classi ying o ally geodesic submani olds in symme ic spaces has
been an ou s anding opic o esea ch in submani old geome y o e he las decades.
This was s a ed by Wol [187] in he six ies, when he classi ied hese objec s in
symme ic spaces o ank one. Fo ank wo, his p oblem has been add essed by
Chen, Nagano [48, 49] and Klein [107, 108, 109]. Indeed, up o da e, we only ha e
comple e classi ica ions in symme ic spaces o ank less o equal han wo.
Any o ally geodesic submani old o a symme ic space is i sel a symme ic space.
E en on an i educible symme ic space, he e can exis educible o ally geodesic
submani olds. Thus, in o de o ha e a comple e classi ica ion o o ally geodesic
submani olds in a gi en i educible symme ic space i is necessa y o ha e a good
unde s anding o o ally geodesic submani olds o educible symme ic spaces.
We ex end Wol ’s esul o p oduc s o ank one symme ic spaces. We will see
ha he o ally geodesic submani olds o p oduc s o ank one symme ic spaces
admi a nice combina o ial desc ip ion. Fi s ly, we in oduce some sligh modi ica ion
o Young ableaux ha we call adap ed Young ableaux (see Sec ion
§
6.2 o he
de ini ion), which will be use ul o classi y o ally geodesic submani olds in a bi a y
p oduc s o symme ic spaces o ank one and o de e mine hei isome y ype. We
10 1 P elimina ies
whe e G·p={g·p:g∈G}deno es he o bi o G h ough p∈Mand he g oup
Gp={g∈G:g·p=p}i s iso opy a p∈M. We will w i e G↷M o deno e he
ac ion o Gon M.
Two isome ic ac ions G1↷M1and G2↷M2a e isomo phic i he e exis s a Lie
g oup isomo phism ψ:G1→G2and an isome y :M1→M2in such a way ha
(gp) = ψ(g) (p), o all p∈Mand g∈G1.
Le M/Gbe he se o o bi s o he isome ic ac ion o he Lie g oup Gon M.
In o de o ensu e ha he o bi s a e closed and hus embedded submani olds o M,
we will equi e ha Gac s p ope ly on M. The ac ion o Gon Mis p ope i he
map G×M→M×M, (g, p)7→ (p, g ·p), is p ope . Since Gac s by isome ies,
his is equi alen o he ac ha Gis (up o e ec i iza ion) a closed subg oup o
Isom(M). See [58] o mo e in o ma ion abou p ope ac ions. A (closed) embedded
submani old o Mis ex insically homogeneous i i is an o bi o a closed subg oup
o Isom(M). F om now on, we will always assume ha isome ic ac ions a e p ope .
Two o bi s G·pand G·q, whe e p, q ∈M, a e equi alen i he iso opy subg oups
Gpand Gqa e conjuga e in G. Le us deno e by [G·p] he equi alence class o G·p. A
pa ial o de ing ≤on he se o equi alence classes o o bi s o Gon Mcan be de ined
by
[G·p]≤[G·q] : ⇐⇒ Gqis conjuga e o a subg oup o Gpin G.
An o bi G·pis called p incipal i [G·p] is maximal o ≤. All p incipal o bi s
a e equi alen , and hen, hey ha e he same codimension in M, which is called he
cohomogenei y o he ac ion o Gon M. Mo eo e , he union o he p incipal o bi s
is an open and dense subse o Mand we can eco e all he nea by o bi s o he
ac ion o Gby knowing a p incipal o bi , see [14,
§
2.1.8]. Indeed, i G·pis a p incipal
o bi o a G-ac ion on Mand F⊂Mis any o he o bi o G, he e exis s a ec o
ξp∈νp(G·p) such ha expp(ξp) = q∈F, which can be ex ended o a G-equi a ian
no mal ec o ield in G·p, since G·pis p incipal, and G·q={expx(ξx) : x∈G·p}. An
o bi G·pis called singula i i s codimension is la ge han he cohomogenei y o he
ac ion, and is called excep ional i i s codimension coincides wi h he cohomogenei y
o Gbu is no p incipal.
Le p∈Mand conside he ac ion o Gpon TpM. The es ic ion o his ac ion
o Tp(G·p) is called he iso opy ep esen a ion o he G-ac ion a p∈M, and he
es ic ion o νp(G·p) is called he slice ep esen a ion a p∈M. Le p∈Mand G
be a connec ed Lie g oup ac ing isome ically and p ope ly on M. A slice a p∈M
is an embedded submani old Spo Mpassing h ough p ha sa is ies:
i) TpM=Tp(G·p)⊕Tp(Sp), and TqM=Tq(G·p) + Tq(Sp), o e e y q∈Sp.
ii) Fo e e y q∈Spand g∈G, we ha e g·q∈Spi and only i g∈Gp.
Mo eo e , his implies ha he e exis s a G-equi a ian di eomo phism be ween G·Sp
and he o al space o he bundle wi h ibe Sp
Sp→G×GpSp→G/Gp,

1.3 Homogeneous and symme ic spaces 11
associa ed wi h he Gp-p incipal bundle desc ibed abo e. The ac ion o Gpon Spis
isomo phic o he slice ep esen a ion es ic ed o an open ball o νp(G·p). Fu he -
mo e, he cohomogenei y o he slice ep esen a ion a e e y poin coincides wi h he
cohomogenei y o he ac ion o Gon M, and he o bi G·pis p incipal i and only
i he slice ep esen a ion a pis i ial. I was p o ed in [155] ha p ope ac ions
always ha e slices a e e y poin o M.
An impo an class o isome ic ac ions is cons i u ed by pola ac ions. A p ope
isome ic ac ion o Gon Mis pola i he e exis s a comple e, closed and embedded
submani old Σ (called sec ion) which in e sec s all he o bi s o he G-ac ion on M
o hogonally. I ollows ha he cohomogenei y o Gcoincides wi h he dimension o
Σ and i can be p o ed ha Σ is a o ally geodesic submani old o M. See [129] o a
de ailed p oo o hese ac s. Mo eo e , i Σ is la , he ac ion is said o be hype pola .
1.3 Homogeneous and symme ic spaces
Homogenei y is a cen al no ion in Ma hema ics. The o igin o homogeneous spaces
da es back o he eme gence o non-Euclidean geome y in he mid-19 h cen u y. The
geome y o hese spaces is qui e di e en om ha o he Euclidean spaces ha we
a e accus omed o s udying in high school. A his poin , he need a ises o cla i y how
o de ine geome y. E langen’s p og am answe s his ques ion. This was p oposed by
Felix Klein in 1872. Basically, geome y was de ined as he s udy o hose p ope ies
in a space ha a e in a ian unde a gi en ans o ma ion g oup.
In ui i ely, a homogeneous space is a space ha looks he same a each poin . Fo
his eason, homogeneous spaces se e as a model space o a ious ypes o geome ic
s uc u es. In pa icula , ou in e es lies in hose homogeneous spaces ha a ise om
isome ic ac ions, ha is, ac ions p ese ing he me ic.
A symme ic space is a homogeneous space whose isome y g oup con ains an
in e sion symme y a each poin . Symme ic spaces a ise in a b oad di e si y o
si ua ions in bo h Ma hema ics and Physics. Thei o igin goes back o he ollowing
ques ion posed by Ca an in 1926:
Which a e he Riemannian mani olds whose cu a u e enso R
is p ese ed by pa allel anspo along any cu e?
This p ope y is equi alen o he equa ion ∇R= 0, and he spaces sa is ying his
p ope y a e in ima ely ela ed o symme ic spaces. Indeed, e e y Riemannian man-
i old sa is ying ∇R= 0 is locally isome ic o a symme ic space. Ca an achie ed a
comple e classi ica ion o symme ic spaces in [43].
1.3.1 Homogeneous spaces
Fo a nice in oduc ion o he heo y o homogeneous spaces, one can consul [6] o
[111, Chap e X]. A Riemannian mani old Mis homogeneous i he e exis s some
subg oup Go Isom(M) such ha Gac s ansi i ely on M. We ix a poin o∈M, so
Mis di eomo phic o G/K, whe e K=Gois he iso opy a o, by he map Φ: G/K→
12 1 P elimina ies
Mde ined by gK→g(o). We pull back he me ic o Mby Φ o G/K, u ning
Φ in o an isome y. Fu he mo e, he me ic ⟨·,·⟩ induced in G/Kis G-in a ian .
Homogeneous spaces a e analy ic Riemannian mani olds, see [35, Lemma 1.1].
Fo any X∈g, whe e gis he Lie algeb a o G, we can associa e a Killing ec o
ield X∗gi en by X∗
p=d
d | =0(Exp( X)·p), o e e y p∈M, whe e Exp deno es
he Lie exponen ial map o G. Homogeneous spaces can be cha ac e ized in e ms o
Killing ec o ields as ollows. A Riemannian mani old Mis homogeneous i o e e y
p∈Mand e e y ∈TpM, he e is a Killing ield X∈Γ(T M) such ha Xp= .
Riemannian homogeneous spaces G/Kalways ha e a educ i e decomposi ion. A
educ i e decomposi ion is a spli ing g=k⊕p, whe e kis he Lie algeb a o Kand
pis an Ad(K)-in a ian subspace o g. Thus, we ha e he b acke ela ions [k,p]⊂p
and [k,k]⊂k. I we conside he linea iza ion a o∈Mo he iso opy ac ion o
Kon M, we ge he iso opy ep esen a ion o he homogeneous space M, which
is de ined as k∈K7→ k∗o∈GL(ToM), whe e k∗odeno es he di e en ial o ka
o∈M. This is equi alen o he adjoin ep esen a ion o G es ic ed o Kon p,
since pand ToMcan be iden i ied by he map which sends X∈p o X∗
o∈ToM.
A homogeneous space M=G/Kis iso opy i educible i he iso opy ep esen a ion
is an i educible ep esen a ion. I M=G/Kis iso opy i educible, Schu ’s lemma
implies ha he e is a unique G-in a ian me ic on Mup o homo he y, and ha M
is an Eins ein mani old. Mo eo e , Mis s ongly iso opy i educible i he es ic ion
o he iso opy ep esen a ion o he connec ed componen o Kis i educible.
Le us deno e by Xkand Xp he p ojec ion o X∈gon o kand p, espec i ely.
We de ine a symme ic bilinea map U:p×p→pby
2⟨U(X, Y ), Z⟩=⟨[Z, X]p, Y ⟩+⟨X, [Z, Y ]p⟩,
whe e X, Y, Z ∈pand ⟨·,·⟩deno es he inne p oduc on pinduced by he Riemannian
me ic on M. The educ i e decomposi ion g=k⊕pis na u ally educ i e i Uis
iden ically ze o. In pa icula , i U≡0, e e y geodesic γo Mpassing h ough o∈M
is gi en by o 7→ Exp( X) wi h X∈p, whe e Exp deno es he Lie exponen ial map o
G. A homogeneous space whe e e e y geodesic is an o bi o a 1-pa ame e subg oup o
he isome y g oup is said o be a geodesic o bi space, o , o sho , g. o. space. Thus,
na u ally educ i e spaces a e g. o. Mo eo e , we say ha he educ i e decomposi ion
g=k⊕pis no mal homogeneous i he e exis s some Ad(G)-in a ian inne p oduc q
on gsuch ha ⟨·,·⟩ =q|p×pand p=k⊥, whe e k⊥deno es he o hogonal complemen
o kin gwi h espec o q. I u ns ou ha e e y no mal homogeneous educ i e
decomposi ion is na u ally educ i e. Indeed, we ha e he chain o s ic inclusions
No mal
homogeneous spaces⊊Na u ally educ i e
homogeneous spaces⊊Geodesic o bi
spaces .
Le us conside he canonical connec ion ∇cassocia ed wi h a educ i e decom-
posi ion g=k⊕p, which is he unique G-in a ian a ine connec ion on Msuch
ha
(∇c
X∗Y∗)o= (−[X, Y ]p)∗
o,(1.1)
1.3.2 Symme ic spaces 13
whe e X, Y ∈p. We can exp ess he Le i-Ci i a connec ion o Ma oas
(∇X∗Y∗)o=−1
2[X, Y ]p+U(X, Y )∗
o
,(1.2)
whe e X, Y ∈p. The di e ence enso Da o∈Mis de ined as D= (∇−∇c)o, and
using he iden i ica ion o ToMand p, we ha e
DXY=1
2[X, Y ]p+U(X, Y ), o e e y X, Y ∈p. (1.3)
Using he o mula o he Le i-Ci i a connec ion and he iden i ica ion o pwi h
ToM, we can compu e he cu a u e enso o M, which is gi en by
Ro(X, Y )Z=1
2[Z, [X, Y ]p]p−[[X, Y ]k, Z]]p−U(Z, [X, Y ]p) + 1
4[[Z, Y ]p, X]p
−1
2U([Z, Y ]p, X)−1
2[U(Z, Y ), X]p+U(U(Z, Y ), X)−1
4[[Z, X]p, Y ]p
+1
2U([Z, X]p, Y ) + 1
2[U(Z, X), Y ]p−U(U(Z, X), Y ),
(1.4)
whe e X, Y, Z ∈p. We end up his sec ion wi h a ema k ha will be use ul o compu e
he co a ian de i a i es o he cu a u e enso a he base poin o∈M=G/K.
Rema k 1.3.1.By [42, P oposi ion 1.4.15] he cu a u e enso Ro a Riemannian
homogeneous space M=G/Ksa is ies ∇cR= 0, since i is G-in a ian . Le g=
k⊕pbe a educ i e decomposi ion o M=G/K. Then, using he de ini ion o he
di e ence enso and he iden i ica ion o pwi h ToMwe ha e
(∇VR)(X, Y, Z) = ((∇V−∇c
V)R)(X, Y, Z)
=DVR(X, Y )Z−R(DVX, Y )Z−R(X, DVY)Z−R(X, Y )DVZ,
whe e X, Y, Z, V ∈p.
1.3.2 Symme ic spaces
Fo a de ailed exposi ion o he heo y o symme ic spaces one can ollow [90], [91],
[127], and [128]. Fo a quicke in oduc ion, we ecommend [77] o [193].
Le Mbe a Riemannian mani old. We say ha Mis a symme ic space i o
e e y poin p∈M he e exis s an isome y sp∈Isom(M) such ha sp ixes p∈M
and s∗p=−IdTpM. The isome y spis called geodesic e lec ion a p∈M.
F om he de ini ion, we can deduce ha symme ic spaces a e comple e, since
geodesics can be ex ended by using geodesic e lec ions. This implies ha o any
p, q ∈M, he e is a geodesic segmen γjoining pand q. Thus, so, whe e ois he
mid-poin o γ, maps p o q, p o ing ha e e y symme ic space is homogeneous. Le
G= Isom0(M) be he connec ed componen o Isom(M) con aining he iden i y and
Kbe he iso opy o Ga o∈M.
14 1 P elimina ies
Symme ic spaces can be cha ac e ized in e ms o Killing ec o ields as ollows.
A Killing ec o ield Xon a Riemannian mani old Msuch ha (∇X)p= 0 is called
a ans ec ion a p∈M. A Riemannian mani old Mis symme ic i and only i o
e e y poin p∈Mand e e y ∈TpM, he e is a ans ec ion X∈Γ(TM) wi h
Xp= .
Now conside σ:G→G, gi en by g∈G7→ sogs−1
o∈G. Then, G0
σ⊂K⊂Gσ,
whe e G0
σis he connec ed componen o Gσ:= {g∈G:σ(g) = g}. The map σis
an in olu i e au omo phism o Lie g oups, and i s di e en ial θ=σ∗e:g→gis an
in olu i e au omo phism o Lie algeb as. The map θis called he Ca an in olu ion
o he symme ic space M=G/Kand i spli s gin o he sum o he eigenspaces o
θ,kand p, associa ed wi h he eigen alues 1 and −1, espec i ely. This p o ides a
educ i e decomposi ion o M=G/Kgi en by g=k⊕p, whe e pis iden i ied wi h
ToM, and kis he Lie algeb a o K. The ank o a symme ic space Mis he dimension
o a maximal la o ally geodesic submani old o M, o equi alen ly, he dimension
o a maximal abelian subspace o p. Fu he mo e, in he case o symme ic spaces
we ha e [p,p] = k. This implies ha U anishes iden ically, p o ing ha symme ic
spaces a e na u ally educ i e homogeneous spaces. Mo eo e , by Equa ion (1.4), he
cu a u e enso o Ma ocan be exp essed as
Ro(X, Y )Z=−[[X, Y ], Z] (1.5)
o X, Y, Z ∈p≃ToM.
Le adX:g→g he map gi en by adX(Y)=[X, Y ]. Now conside Bgbe he
Killing o m o g, ha is,
Bg(X, Y ) = (adX◦adY) o X, Y ∈g.
I ollows ha Bg(X, Y ) = 0, o e e y X∈kand Y∈p. I he Lie algeb a o which
we conside he Killing o m is clea om he con ex , we will simply w i e B.
A symme ic space M=G/Kis said o be o compac ype, o non-compac ype
o o Euclidean ype i B|p×p, he es ic ion o B o p, is nega i e de ini e, posi i e
de ini e o iden ically ze o, espec i ely. I M=G/Kis iso opy i educible, Schu ’s
lemma yields ha B|p×pis a mul iple o he induced me ic on p≃ToM. Hence, i
Mis iso opy i educible, he ype is a mu ually exclusi e p ope y o M. Le
Mbe
he uni e sal co e ing o M. Then,
Mis again a symme ic space and by De-Rham
Theo em,
M=
M0×
M1×···×
Mk, whe e
M0is isome ic o a Euclidean space and
Miis a simply connec ed i educible symme ic space, wi h i∈ {1, . . . , k}. We say
ha Mis semisimple i
M0is jus a poin . In his case he Lie algeb a gis semisimple.
Mo eo e , i Mis semisimple, Mis i educible i and only i i is iso opy i educible.
An impo an no ion, which es ablishes a ela ion be ween symme ic spaces o
compac ype and non-compac ype, is duali y. I we es ic ou a en ion o simply
connec ed symme ic spaces, he e is a one- o-one co espondence be ween symme ic
spaces o non-compac ype and symme ic spaces o compac ype. A he Lie algeb a
le el his wo ks as ollows. Le M=G/Kbe a symme ic space o non-compac ype
and le g=k⊕pbe he educ i e decomposi ion induced by he Ca an in olu ion θ.
Conside gC=g⊗RC, he complexi ica ion o g. We can de ine he subspace g∗=k⊕ip
1.3.3 Symme ic spaces o non-compac ype 15
o gC, whe e i=√−1. Then g∗is a compac Lie algeb a and M∗=G∗/K∗is a
symme ic space o compac ype equipped wi h he Riemannian me ic induced by
he nega i e o he Killing o m o g∗, whe e G∗is he simply connec ed Lie g oup
wi h Lie algeb a g∗and K∗is he connec ed subg oup o G∗wi h Lie algeb a k.
1.3.3 Symme ic spaces o non-compac ype
The symme ic spaces o non-compac ype a e o pa icula ele ance o his hesis
since many esul s ha we ob ain a e p o ed in his se ing. See [65] o mo e de ails.
Le M=G/Kbe a symme ic space o non-compac ype and conside he Ca an
in olu ion θo ginduced by he geodesic symme y a he base poin o∈M. The
educ i e decomposi ion g=k⊕p, induced by θ, is called he Ca an decomposi ion
o g. Le us conside he posi i e de ini e inne p oduc on ggi en by
Bθ(X, Y ) = −B(θX, Y ) o e e y X, Y ∈g.
A use ul ac abou his inne p oduc is ha he adjoin map o adX:g→gwi h
espec o Bθis −adθX o e e y X∈g.
The iso opy ep esen a ion o Ma ois pola and e e y maximal abelian subspace
o pis a sec ion o his ac ion. Thus, wo maximal abelian subspaces o pa e
conjuga e by an elemen o K. Le abe a maximal abelian subspace o p. Mo eo e ,
i can be p o ed ha Mis simply connec ed and hus i is di eomo phic o a Euclidean
space, since i has non-posi i e sec ional cu a u e.
Since a⊂p, e e y ope a o adH:g→gis sel -adjoin wi h espec o Bθ. Mo e-
o e , since [adH1,adH2] = ad[H1, H2] = 0, he se {adH:H∈a}cons i u es a
commu ing amily o sel -adjoin endomo phisms o g. Thus, hey diagonalize simul-
aneously. Thei common eigenspaces a e he ( es ic ed) oo spaces o gand he
non-ze o eigen alues (which depend linea ly on H∈a) a e he ( es ic ed) oo s o g.
Fo each λ∈a∗, we de ine
gλ={X∈g: adHX=λ(H)X o all H∈a}.
Then, any λ= 0 such ha gλ= 0 is a oo and e e y gλ= 0 is a oo space. I can
be checked ha
[gλ,gµ]⊂gλ+µ o e e y λ, µ ∈a∗.
Le ∆ deno e he se o oo s. Then we ha e he ollowing o hogonal decomposi-
ion wi h espec o Bθ:
g=g0⊕ M
λ∈∆
gλ!,
which is called he ( es ic ed) oo space decomposi ion o g. We ha e θgλ=g−λ,
implying ha λ∈∆ i and only i −λ∈∆. Addi ionally, g0=k0⊕a, whe e k0=g0∩k
is he no malize o ain k.
Fo each λ∈∆, we de ine Hλ∈aas he unique elemen o asa is ying B(Hλ, H) =
λ(H), o all H∈a. This induces an inne p oduc on a∗gi en by ⟨λ, µ⟩=B(Hλ, Hµ),
o e e y λ, µ ∈a∗. Mo eo e , i can be p o ed ha ∆ de ines a oo sys em in a∗,
hus sa is ying:

16 1 P elimina ies
i) a∗is spanned by ∆,
ii) nα,β = 2⟨α, β⟩/⟨α, α⟩ ∈ Z,
iii) β−nα,βα∈∆, o e e y α, β ∈∆.
Now choose a hype plane in a∗such ha i does no con ain any oo . We can
de ine a posi i i y c i e ion on ∆ by decla ing hose oo s ha lie a one o he wo
hal -spaces de e mined by he hype plane o be posi i e. I ∆+deno es he se o
posi i e oo s, hen ∆ = ∆+∪(−∆+). Fu he mo e, we can de ine he se o simple
oo s Π as he subse o hose posi i e oo s which canno be exp essed as he sum o
wo posi i e oo s. The subspace
n=M
λ∈∆+
gλ
o gis a nilpo en subalgeb a o gand a⊕nis hen a sol able subalgeb a such ha
[a⊕n,a⊕n] = n. Any wo choices o posi i e c i e ia on ∆ gi e ise o nilpo en
subalgeb as nwhich a e conjuga e by an elemen o he g oup NK(a) = {k∈K:
Ad(k)a⊂a}.
The Iwasawa decomposi ion heo em (see [98]) s a es ha
g=k⊕a⊕n
is a ec o space di ec sum. Obse e ha his sum is nei he an o hogonal sum no
a semidi ec sum. Le Aand Nbe he connec ed Lie subg oups o Gwi h Lie algeb as
aand n, espec i ely. The connec ed Lie subg oup o Gwi h Lie algeb a a⊕nis a
semidi ec p oduc AN, since [a,n]⊂n. Then, he Iwasawa heo em a he Lie g oup
le el s a es ha he mul iplica ion map
K×A×N→G,(k, a, n)7→ kan,
is an analy ic di eomo phism. Mo eo e , he Lie g oups Aand Na e simply con-
nec ed, and hus hey and AN a e di eomo phic o Euclidean spaces. The smoo h
map Φ|AN →Mis a di eomo phism. This allows us o pull back he Riemannian
me ic on M o AN. Mo eo e , his me ic on AN is le -in a ian . Consequen ly,
e e y symme ic space M=G/Ko non-compac ype is isome ic o a sol able Lie
g oup AN equipped wi h a le -in a ian me ic. In pa icula , his shows ha Mis
di eomo phic o a Euclidean space. By Equa ion (1.5), Mis non-posi i ely cu ed,
and hus Mis a Hadama d mani old.
A use ul concep ela ed o a symme ic space o non-compac ype Mis ha o
ideal bounda y. The ideal bounda y M(∞) o Mis de ined as he se o equi alence
classes o comple e, uni -speed geodesics o Munde he ela ion
γ1∼γ2:⇔ {d(γ1( ), γ2( )) : ≥0}is bounded.
Now, we can in oduce wi h he so-called cone opology on M⊔M(∞), see [75] o
mo e de ails, in such a way ha M⊔M(∞) becomes homeomo phic o a Euclidean
closed ball, whe e Mco esponds o i s in e io and M(∞) o i s bounda y. Finally,
i is impo an o no ice ha he ac ion o Gon Mcan be na u ally ex ended o
M(∞) by aking g·[γ] := [g·γ].
1.4 Heisenbe g algeb as and hype bolic spaces 17
1.4 Heisenbe g algeb as and hype bolic spaces
Gene alized Heisenbe g algeb as a e highly signi ican o his hesis since hey a e
closely ela ed o symme ic spaces o ank one. In pa icula , symme ic spaces o
ank one and o non-compac ype cons i u e a special case o Damek-Ricci spaces,
which a e sol able Lie g oups equipped wi h a le -in a ian me ic whose Lie al-
geb as a e ob ained as ce ain one-dimensional ex ensions o gene alized Heisenbe g
algeb as. I u ns ou ha his s uc u e is pa icula ly well-sui ed and ele an o
s udy submani old geome y in hese spaces as was shown in [13, 60] o [70].
1.4.1 Cli o d algeb as
In his subsec ion, we ix some no a ion and ecall ce ain well-known ac s ela ed o
Cli o d algeb as. We will mainly ollow [120]. Le us s a by in oducing he no ion
o Cli o d algeb a. Le Vbe a eal ec o space o e and qbe a quad a ic o m on
V. Le T(V) := L∞
=0 T (V) be he enso algeb a o V, whe e T (V) := V⊗ )
. . . ⊗V
and T0(V) = R. This is an associa i e, uni a y and g aded algeb a whe e Tk(V) is
cons i u ed by he homogeneous elemen s o deg ee k∈N. Le Tq(V) be he wo-sided
ideal in T(V) gene a ed by all elemen s o he o m ⊗ +q( )1, whe e ∈V. We
de ine Cl(V, q), he Cli o d algeb a associa ed wi h Vand q, as he quo ien algeb a
Cl(V, q) := T(V)/Tq(V).
Le Vand V′be wo ec o spaces equipped wi h quad a ic o ms qand q′. Then,
e e y linea map : (V, q)→(V′, q′) such ha (q( )) = q′( ( )) o e e y ∈V,
induces a mo phism be ween Clq(V) and Clq′(V′) in he na u al way.
Two Cli o d algeb as Cl(V, q) and Cl(V′, q′) wi h dim V= dim V′and such ha
qand q′ha e he same signa u e a e isomo phic. Since we will conside only Cli o d
algeb as whe e qis a posi i e de ini e quad a ic o m, in o de o simpli y ou no a ion
we will w i e Clno Cl(V) ins ead o Cl(V, q), whe e Vhas dimension n.
Le Fbe he no med di ision algeb a o he eal numbe s R, he complex numbe s
Co he qua e nions H. Deno e by F(k) he algeb a o ma ices o o de kwhose
en ies a e in F. In Table 1.1, we lis Cli o d algeb as Cln, whe e n≤8. No ice ha
one has he pe iodici y isomo phism Cln+8 ≃Cln⊗Cl8. Hence, i is enough o lis
Cln, wi h n≤8, o de e mine Cln o e e y n∈N.
n1 2 3 4 5 6 7 8
ClnCHH⊕H H(2) C(4) R(8) R(8) ⊕R(8) R(16)
Table 1.1: Cli o d algeb as Cln o n≤8.
I can be p o ed ha e e y i educible ep esen a ion o F(k) is equi alen o he
s anda d ac ion on Fk, and ha F(k)⊕F(k) has exac ly wo equi alence classes o
i educible ep esen a ions, gi en by he s anda d ac ion o each one o he wo ac o s
on Fk, see [120, Theo em 5.6].
18 1 P elimina ies
Mo eo e , Clnis isomo phic ei he o F(k), when n≡ 3 mod 4, o o F(k)⊕
F(k), when n≡3 mod 4. As men ioned abo e, he e exis s a unique i educible
ep esen a ion ρin he i s case and exac ly wo i educible ep esen a ions ρ+and
ρ−in he second case, up o equi alence. In o de o dis inguish be ween ρ+and ρ−,
we in oduce he olume elemen o Cln. Le us ix an o ien a ion (e1, . . . , en) in V.
Then we de ine ω=e1···en∈Cl(V) as he olume elemen o Cln. I u ns ou ha
ρ+(ω) = Id and ρ−(ω) = −Id, when n≡3 mod 4.
1.4.2 Gene alized Heisenbe g algeb as
In wha ollows, we will in oduce he basic concep s needed o de ine gene alized
Heisenbe g algeb as. See [30] o a nice and comple e su ey on his opic.
Le us conside wo non-ze o eal ec o spaces and z, and β: × →za
skew-symme ic bilinea map. We de ine n:= ⊕zand we endow i wi h an inne
p oduc ⟨·,·⟩ such ha and za e o hogonal. Mo eo e , we in oduce a linea map
J:Z∈z7→ JZ∈End( ) gi en by
⟨JZU, V ⟩=⟨β(U, V ), Z⟩, o all U, V ∈ , Z ∈z,
and we de ine a Lie b acke in nby
[U+X, V +Y] = β(U, V ), o all U, V ∈ , X, Y ∈z.
Then, nis a wo-s ep nilpo en Lie algeb a whose cen e is Z(n) = z. I , in addi ion
o ha , we ha e J2
Z=−⟨Z, Z⟩id o e e y Z∈z, hen nis said o be a gene al-
ized Heisenbe g algeb a, and he associa ed simply connec ed nilpo en Lie g oup N,
endowed wi h he induced le -in a ian Riemannian me ic, is called a gene alized
Heisenbe g g oup. The mo e classical no ions o Heisenbe g algeb as and g oups a e
eco e ed p ecisely when zis one-dimensional.
Le U,V∈ and X,Y∈z. One has he ollowing well-known p ope ies o
gene alized Heisenbe g algeb as (see [30, Chap e 3]):
JXJY+JYJX=−2⟨X, Y ⟩id ,[JXU, V ]−[U, JXV] = −2⟨U, V ⟩X,
⟨JXU, JXV⟩=⟨X, X⟩⟨U, V ⟩,⟨JXU, JYU⟩=⟨X, Y ⟩⟨U, U⟩.
In pa icula , o any uni Z∈z,JZis a complex s uc u e on . Mo eo e , he
map J:z→End( ) can be ex ended o he Cli o d algeb a Cl(z, q), whe e qis he
quad a ic o m induced by ⟨·,·⟩, in such a way ha becomes a Cli o d module o e
Cl(z, q).
1.4.3 Symme ic spaces o ank one and non-compac ype
Hu wi z’s heo em asse s ha any no med eal di ision algeb a Fis isomo phic o
R,C,Ho O. The hype bolic spaces o e hese algeb as cons i u e he symme ic
spaces o non-compac ype and ank one. In o he wo ds, i Mis a symme ic space
o non-compac ype and ank one, hen Mis ei he a eal hype bolic space RHn+1,
1.4.3 Symme ic spaces o ank one and non-compac ype 19
n≥1, a complex hype bolic space CHn+1,n≥1, a qua e nionic hype bolic space
HHn+1,n≥1, o he Cayley hype bolic plane OH2. As a symme ic space, any o
hese mani olds Mcan be iden i ied wi h a quo ien G/Ko Lie g oups, whe e Gis
he connec ed componen o he iden i y o he isome y g oup o M, up o a ini e
co e ing, and Kis he iso opy subg oup o Gco esponding o a ce ain poin o∈M
ha we ix om now on. Then one can ake G=SO0
1,n+1,SU1,n+1,Sp1,n+1,F−20
4
and K=SOn+1,S(U1×Un+1), Sp1×Spn+1,Spin9, depending on whe he F=R,C,
H,O, espec i ely.
We deno e by gand k he Lie algeb as o Gand K, espec i ely, by B he Killing
o m o g, and by θ he Ca an in olu ion o gwi h espec o k. Le g=k⊕pbe he
Ca an decomposi ion o ginduced by θ. We ha e ha ⟨X, Y ⟩=−B(X, θY ) is an
inne p oduc ha es ic ed o pinduces a Riemannian me ic on G/K ha makes
G/Kisome ic o M, up o homo he y.
RHn+1 CHn+1 HHn+1 OH2
G SO0
1,n+1 SU1,n+1 Sp1,n+1 F−20
4
K SOn+1 S(U1Un+1)Sp1Spn+1 Spin9
K0SOnS(U1Un)Sp1SpnSpin7
gαRnCnHnO
g2α0R R3R7
Table 1.2: Da a o each hype bolic space.
Le abe a maximal abelian subspace o p, which is one-dimensional as Mhas ank
one. Then, he co esponding oo space decomposi ion o gadop s he o m
g=g−2α⊕g−α⊕g0⊕gα⊕g2α.
He e, he oo space g0spli s as g0=k0⊕a, whe e k0is he Lie algeb a o
K0=NK(a), he no malize o ain K, which also no malizes gαand cen alizes g2α.
Mo eo e , g=k⊕a⊕n, whe e n=gα⊕g2α, is an Iwasawa decomposi ion o g.
When F=R, we ha e g−2α=g2α= 0 and nis abelian. O he wise, nis only wo-s ep
nilpo en . In ac , nis isomo phic o he (2n+ 1)-dimensional Heisenbe g algeb a
when F=Cand o a ce ain gene alized Heisenbe g algeb a i F∈ {H,O}. Mo eo e ,
g2α, he cen e o n, is equal o he de i ed algeb a o n, and has dimension 1, 3 o 7
o F=C,Ho O, espec i ely.
In addi ion o his, we can iden i y gαwi h Rn,Cn,Hn,O o F=R,C,H,O,
espec i ely. Indeed, gαis a Cli o d module o e Clm, whe e m= dim g2α, which
is he sum o equi alen Cli o d modules i m= 3, and is i educible i m= 7.
The possibili ies o G,K,K0and he oo spaces co esponding o posi i e oo s a e
summa ized in Table 1.2.
26 2 A non-isopa ame ic hype su ace wi h cons an p incipal cu a u es
i ) a ho osphe e o RHn.
An impo an consequence o hese classi ica ions is ha e e y isopa ame ic hy-
pe su ace in Rno RHnis an open subse o a homogeneous one. Thus, hese
heo ems p o ide he classi ica ions o homogeneous hype su aces in hese spaces.
Hence, he ela ionship be ween he h ee p ope ies de ined abo e in hese wo am-
bien spaces is locally he one ou lined in Figu e 2.2.
Figu e 2.2: Hype su aces in Rnand RHn.
The p oblem in ound sphe es u ned ou o be much mo e complica ed. Ca an
[44] classi ied hype su aces wi h g∈ {1,2,3}cons an p incipal cu a u es, and wi h
g= 4 i all he mul iplici ies a e simple. Howe e , he was no able o sol e he
gene al case. La e on, M¨unzne de eloped he heo y o Ca an u he and p o ed
in [139, 140] ha he numbe o dis inc p incipal cu a u es o an isopa ame ic
hype su ace mus be g∈ {1,2,3,4,6}. Howe e , he e is a main di icul y in he
p oblem o classi iying isopa ame ic hype su aces in Sn: no e e y isopa ame ic
hype su ace is homogeneous. Su p isingly, in [153], some inhomogeneous examples
wi h g= 4 we e ound.
Figu e 2.3: Hype su aces in Sn.
In 2007-2008, Cecil, Chi, Jensen [45] and Imme oll [97] made subs an ial p og ess
in he classi ica ion o isopa ame ic hype su aces wi h g= 4 dis inc p incipal cu -
a u es. La e , Chi concluded he case g= 4 in a se ies o a icles [50, 51, 52]. The
las case, g= 6, occu s only in S7and S13, see [1]. In S7such hype su aces a e
homogeneous and hey a e classi ied [72, 162]. Miyaoka [134] deal wi h he p ob-
lem in S13, bu , as i was explained by Si e in [162, 163], he e seems o be an
issue in his a icle and in he pos e io e a um ha Miyaoka [135] w o e yielding
ce ain con o e sy. Howe e , i is belie ed ha e e y isopa ame ic hype su ace
wi h g∈ {1,2,3,6}is homogeneous. We will desc ibe homogeneous hype su aces in
sphe es in Chap e 3.

2.2 Isopa ame ici y and cons an p incipal cu a u es in ank one 27
Le us desc ibe he inhomogeneous isopa ame ic hype su aces wi h g= 4 dis inc
p incipal cu a u es in ound sphe es. Le V=R2n+2 be a Euclidean space. We say
ha an (m+1)- uple (P0, . . . , Pm) o eal sel -adjoin endomo phisms o Vis a Cli o d
sys em in End(V) i i sa is ies:
PiPj+PjPi= 2δijId,
o all i, j ∈ {0, . . . , m}, whe e δij is he K onecke del a. Le Pbe he linea span
o a gi en Cli o d sys em and endow i wi h he inne p oduc gi en by ⟨P, P ′⟩=
1
dim(V) (PP′) o P, P ′∈ P. Assume ha n−m > 0. Then, he FKM olia ion FP
associa ed wi h he Cli o d sys em (P0, . . . , Pm) is de ined by he le el se s o F|S(V),
whe e S(V) deno es he uni sphe e o V, and F:V→Ris he polynomial:
F(x) = ⟨x, x⟩2−2
m
X
i=0⟨Pix, x⟩2.
By combining mul iple esul s in [45, 50, 51, 52, 97, 153, 154, 167, 173], we ha e he
ollowing:
Theo em 2.2.3. Le Mbe an isopa ame ic hype su ace o Sn−1⊂Rnwi h g= 4
dis inc p incipal cu a u es. Then, Mis an open pa o a homogeneous hype su ace
o Sn−1o o a egula lea o an FKM olia ion.
Fo spaces wi h non-cons an sec ional cu a u e, he equi alence be ween isopa a-
me ici y and cons ancy o he p incipal cu a u es is no longe ue. In pa icula , i
makes sense o conside some o he Riemannian mani olds wi h non-cons an sec ional
cu a u e and simples cu a u e enso such as hype bolic and p ojec i e spaces o e
a no med di ision algeb a F∈ {C,H,O}. In some o hese spaces, we know he exis-
ence o isopa ame ic hype su aces ha do no ha e cons an p incipal cu a u es.
Fo example, in complex p ojec i e spaces, Wang [181] p o ed he ollowing cha ac-
e iza ion o isopa ame ic hype su aces wi h cons an p incipal cu a u es:
Theo em 2.2.4. Le Mbe an isopa ame ic hype su ace in CPnwi h uni no mal
ec o ield ξ∈Γ(νM). Then, he ollowing a e equi alen :
i) Mhas cons an p incipal cu a u es.
ii) Jξ is a p incipal di ec ion, ha is, Mis a Hop eal hype su ace.
iii) One ocal se o Mis a complex submani old.
In o de o p o ide his example o an isopa ame ic hype su ace wi h non-cons an
p incipal cu a u es, Wang [181] ook an inhomogeneous hype su ace in he sphe e
wi h g= 4 dis inc cons an p incipal cu a u es in S8n+7 ⊂C4(n+1), wi h n≥1, and
p o ed ha i s image unde he Hop ib a ion π:S8n+7 →CP4n+3 does no ha e
complex ocal se s.
Fu he mo e, we know classi ica ions o isopa ame ic hype su aces in CPn, wi h
n= 15 [68], in HPn, wi h n= 7 [69], and CHn, see [64]. I is also known ha
28 2 A non-isopa ame ic hype su ace wi h cons an p incipal cu a u es
Figu e 2.4: Hype su aces in CPnand CHn.
an isopa ame ic hype su ace wi h cons an p incipal cu a u es in CPno CHn
is homogeneous. In he i s case his ollows by combining Theo em 2.2.4 wi h
he classi ica ion o Hop eal hype su aces wi h cons an p incipal cu a u es in
CPn[106], and in he second case i ollows by he classi ica ion in [64].
In HHno OH2 he e a e examples o inhomogeneous hype su aces ha a e
isopa ame ic and ha e cons an p incipal cu a u es, see Sec ion
§
4.5 and [60], e-
spec i ely.
Figu e 2.5: Hype su aces in HPn.
To sum up, he known ela ions be ween he h ee concep s (homogenei y, isopa a-
me ici y and cons ancy o he p incipal cu a u es) o hype su aces in symme ic
spaces o ank one a e explained in Figu es 2.2 o 2.6. In OP2, no ela ion is ye
known apa om he ac ha e e y homogeneous hype su ace is isopa ame ic and
has cons an p incipal cu a u es, which holds o e e y ambien space.
Figu e 2.6: Hype su aces in HHnand OH2.
2.3 A non-isopa ame ic hype su ace wi h cons an p incipal cu a u es 29
Fu he mo e, in Tables 2.1 and 2.2, we summa ize he cu en p og ess in he
classi ica ion o isopa ame ic hype su aces in symme ic spaces o ank one. The
classi ica ion p oblem o homogeneous hype su aces will be discussed in de ail in
Chap e 3. Finally, i is wo h men ioning ha he p oblem o classi ying hype su -
aces wi h cons an p incipal cu a u es in symme ic spaces o ank one seems o be
eally ha d, and we only ha e classi ica ions i we assume ha he numbe o dis inc
p incipal cu a u es is g≤3, in CPn(see [170, 171]) o in CHn(see [15, 16]), o i we
impose some o he hypo heses (see [11, 59, 106, 158]). Fo he sake o b e i y, we will
w i e c.p.c. ins ead o cons an p incipal cu a u es in he ables below, whe e he
ick ( espec i ely, a condi ion on no g) means ha a comple e ( espec i ely, pa ial)
classi ica ion has been ob ained.
SnCPnHPnOP2
Homogeneous ✓ ✓ ✓ ✓
c.p.c. n= 13 g≤3 ? ?
Isopa ame ic n= 13 n= 15 n= 7 ?
Isopa ame ic + c.p.c. n= 13 ✓? ?
Table 2.1: Cu en p og ess in he classi ica ion o hype su aces in symme ic spaces
o compac ype and ank one.
RHnCHnHHnOH2
Homogeneous ✓ ✓ ✓ ✓
c.p.c. ✓g≤3 ? ?
Isopa ame ic ✓ ✓ ? ?
Isopa ame ic + c.p.c. ✓ ✓ ? ?
Table 2.2: Cu en p og ess in he classi ica ion o hype su aces in symme ic spaces
o non-compac ype and ank one.
2.3 A non-isopa ame ic hype su ace wi h cons an
p incipal cu a u es
In his sec ion we cons uc a con o mally la me ic in Rn ha admi s a (non-
Riemannian) olia ion by o ally geodesic, non-isopa ame ic hype planes. Mo eo e ,
he me ic and he olia ion descend o he n-dimensional o us Tn. This p o ides an
example o a non-isopa ame ic hype su ace wi h cons an p incipal cu a u es in a
Riemannian mani old. Also, i shows ha he equi alence be ween isopa ame ici y
and cons ancy o he p incipal cu a u es in spaces o cons an cu a u e does no
hold in he mo e gene al se ing o con o mally la spaces.
30 2 A non-isopa ame ic hype su ace wi h cons an p incipal cu a u es
In o de o ind such a me ic, we need he isome y g oup o be su icien ly small
o spoil he good beha io o pa allel hype su aces. Indeed, i a con o mally la
space admi s a ansi i e g oup o isome ies, hen i is locally symme ic [172], which
would lead us o he appa en ly ou s anding p oblem o inding such an example in
he con ex o symme ic spaces [65,
§
6]. On he o he hand, we cons uc he me ic
so ha i s isome y g oup is no oo small so as o compu e some geodesics explici ly.
2.3.1 The ambien mani old
Le (x1, . . . , xn) deno e he usual coo dina es in Rnand (∂1, . . . , ∂n) he associa ed
coo dina e ec o ields. Fo each n≥2 we de ine a me ic
gij(x1, . . . , xn) := h2(x1, . . . , xn)δij ,
whe e δij is he K onecke ’s del a and
h(x1, . . . , xn) :=
n−1
Y
i=1
(2 + cos(πxi)) ∈R, o each (x1, . . . , xn)∈Rn.
Clea ly, gis con o mally la . We will deno e Rnequipped wi h he me ic gby ¯
Mn.
-4-2 0 2 4
-4
-2
0
2
4
Figu e 2.7: Le el se s o hon R2.
Rema k 2.3.1.In pa icula gis in a ian unde ansla ions o he la ice 2Zn. Hence,
ou me ic gdescends o he o us Tn=Rn/(2Zn).
2.3.2 Ch is o el symbols o ¯
Mn
I is known ha Ch is o el symbols a e gi en by
Γk
ij =1
2gklgjl,i +gli,j −gij,l,
2.3.3 Some e ical geodesics 31
o i, j, k ∈ {1, . . . , n}, whe e we a e using Eins ein summa ion con en ion and we
ha e deno ed he pa ial de i a i e wi h espec o xiby ,i. Thus,
Γk
ij =δjk
2h2h2
,i +δki
2h2h2
,j −δij
2h2h2
,k.
Now o n≥2 we ha e
Γi
ii = (δin −1) πsin(πxi)
2 + cos(πxi),Γk
ij = 0,(2.2a)
Γi
ij = (δjn −1) πsin(πxj)
2 + cos(πxj),Γk
ii = (1 −δkn)πsin(πxk)
2 + cos(πxk),(2.2b)
o mu ually dis inc i, j, k ∈ {1, . . . , n}.
2.3.3 Some e ical geodesics o ¯
Mn
Le us de ine
Ω := {(a1, . . . , an−1, xn)∈Rn:ai∈Z,0≤i≤n−1}.
Le a= (a1, . . . , an−1, xn)∈Ω and γabe he uni -speed geodesic s a ing a awi h
ini ial di ec ion ∂n. By he de ini ion o g, he ollowing maps a e isome ies o ¯
Mn
o each i= 1, . . . , n:

Λi: (x1, . . . , xi, . . . , xn)∈Rn7→ (x1, . . . , −xi, . . . , xn)∈Rn,

Ψi: (x1, . . . , xi, . . . , xn)∈Rn7→ (x1, . . . , xi+ 2, . . . , xn)∈Rn.
Now, o each i∈ {1, . . . , n −1}, we conside he isome y Ψai
i◦Λi. Then, we ha e
ha eγa( ) := Ψai
i◦Λi(γa( )) is ano he geodesic gi en by
eγa( )=(γ1
a( ),...,−γi
a( ) + 2ai, . . . γn
a( )).
Bu eγa( ) and γa( ) ha e he same ini ial condi ions. Hence, by uniqueness we ha e
ha γi
a( ) = ai o each 1 ≤i≤n−1. Obse e ha h(a1, . . . , an−1, x) = 3ρ o any
x∈R, whe e ρis he numbe o e en en ies o (a1, . . . , an−1). Thus, since γa( ) is
pa ame ized by a c leng h we ge ha
γa( )=(a1, . . . , an−1, xn+ 3−ρ ).(2.3)
2.3.4 The Jacobi ope a o
I is clea ha {∂i}n
i=1 is an o hogonal global ame o ¯
Mn. We will compu e ¯
R∂n,
he Jacobi ope a o associa ed wi h ∂n.
All we ha e o do is o compu e he en ies ¯
Rinnj o he cu a u e enso ¯
R o
each i, j ∈ {1, . . . , n}. I i=no j=n, hen ¯
Rinnj = 0. I i, j =n, hen
¯
Rinnj =⟨¯
∇∂i¯
∇∂n∂n, ∂j⟩−⟨¯
∇∂n¯
∇∂i∂n, ∂j⟩−⟨¯
∇[∂i,∂n]∂n, ∂j⟩.

32 2 A non-isopa ame ic hype su ace wi h cons an p incipal cu a u es
On he one hand
⟨¯
∇∂i¯
∇∂n∂n, ∂j⟩=⟨¯
∇∂i(Γk
nn∂k), ∂j⟩=⟨Γk
nn,i∂k+ Γk
nnΓl
ik∂l, ∂j⟩=h2(Γj
nn,i + Γk
nnΓj
ik),
and on he o he hand
⟨¯
∇∂n¯
∇∂i∂n, ∂j⟩=⟨¯
∇∂n(Γk
in∂k), ∂j⟩=⟨Γk
in,n∂k+ Γk
inΓl
nk∂l, ∂j⟩=h2(Γj
in,n + Γk
inΓj
nk).
Since [∂i, ∂j] = 0, we conclude
(¯
R∂n)ij =h2(Γj
nn,i −Γj
in,n + Γk
nnΓj
ik −Γk
inΓj
nk),i i, j =n
0,in any o he case. (2.4)
2.3.5 The example
Le F={Fs}s∈R, whe e Fs={(x1, . . . , xn)∈Rn:xn=s}, o each s∈R. I is
clea ha Fis a olia ion o codimension one on ¯
Mn. Le S,Hand νFsdeno e he
shape ope a o , he mean cu a u e and he no mal bundle o Fs, espec i ely. Then,
each lea is o ally geodesic since ∂n∈Γ(νFs), and using (2.2a) and (2.2b), we ha e
ha ⟨S∂n∂i, ∂j⟩=−⟨¯
∇∂i∂n, ∂j⟩=−h2Γj
in = 0, o each i, j = 1, . . . , n −1.
Rema k 2.3.2.Again, since Fis in a ian by he ac ion o 2Zn, his olia ion descends
o he o us Tn.
Gi en any s∈R, le us conside p∈ Fs, a uni -speed geodesic γ: [0, ε)→¯
Mn
wi h γ(0) = pand ˙γ(0) ∈νpFs o some ε > 0, and M he pa allel hype su ace o
Ma dis ance > 0 sa is ying γ( )∈M . By he Ricca i equa ion (c . [86, Equa ion
3.8]), we ha e
d
d S
˙γ( )=¯
R˙γ( )+ (S
˙γ( ))2,S0
˙γ(0) =S∂n,
whe e S
˙γ( )is he shape ope a o o M a γ( ) wi h espec o he no mal ec o
˙γ( ). Now we ake he ace, so
d
d H
˙γ( )= Ric(˙γ( ),˙γ( )) + ||S
˙γ( )||2,H0
˙γ(0) =H,(2.5)
whe e H
˙γ( )deno es he mean cu a u e o M a γ( ), Ric is he Ricci enso o ¯
Mn
and ||·|| he Hilbe –Schmid no m o an ope a o .
Now we p o e ha no lea o Fis isopa ame ic. Le us conside a∈ Fs∩Ω o
some s∈R. Fi s no e ha ˙γa= 3−ρ∂nby (2.3). By (2.2a) and (2.2b), Γk
ij(γa( )) = 0
and Γn
in = 0 . Hence, by (2.4), we ha e
Ric(˙γa( ),˙γa( )) =
n−1
X
i=1
Γi
nn,i(γa( )) = π2(1 −n+4
3ρ),
whe e we ecall ha ρis he numbe o e en en ies o (a1, . . . , an−1).
2.3.5 The example 33
As a consequence, i a= (0, . . . , 0, s)∈ Fs∩Ω and b= (1, . . . , 1, s)∈ Fs∩Ω,
Ric(˙γa( ),˙γa( )) = n−1
3π2>0 and Ric(˙γb( ),˙γb( )) = (1 −n)π2<0,
o any ∈R.
Bu in ou case, o = 0, we ha e ||S˙γa(0)||2=||S˙γb(0)||2= 0. The e o e, by (2.5),
we deduce ha d
d | =0H
˙γa( )>0 and d
d | =0H
˙γb( )<0. This way we can conclude
ha , o small > 0, he mean cu a u e o he pa allel hype su ace o Fsa dis ance
> 0 is no cons an . Then, Fsis no isopa ame ic.
Chap e 3
Homogeneous hype su aces in
symme ic spaces
In Chap e 2 we ecalled he de ini ion o (ex insically) homogeneous hype su ace
and how hese hype su aces a e ela ed o isopa ame ic hype su aces and hype -
su aces wi h cons an p incipal cu a u es.
The aim o his chap e is o desc ibe he known classi ica ion esul s o homo-
geneous hype su aces in symme ic spaces wi h a special emphasis on hose o ank
one.
This chap e is o ganized in he ollowing way. In Sec ion
§
3.1, we mo i a e
he s udy and ecall some well-known ac s abou cohomogenei y one ac ions. A e
es ablishing he ela ionship be ween cohomogenei y one ac ions and homogeneous
hype su aces, we dedica e Sec ion
§
3.2 o e ise he classi ica ion o homogeneous
hype su aces in symme ic spaces o compac ype, speci ically ocusing on he ank
one case. Then, in Sec ion
§
3.3, we ecall he no ion o gene alized K¨ahle angle,
which will be o eno mous ele ance o he heo y o cohomogenei y one ac ions
on symme ic spaces o non-compac ype and ank one. This will be he opic o
discussion o Sec ion
§
3.4. Finally, Sec ion
§
3.5 is de o ed o explaining he p og am
de eloped by Be nd and Tama u o classi y cohomogenei y one ac ions on symme ic
spaces o non-compac ype and highe ank.
3.1 Cohomogenei y one ac ions
The discipline o geome ic analysis uses he ools om he heo y o pa ial di e en-
ial equa ions (PDEs) o es ablish new esul s in di e en ial geome y. This is due o
he ac ha many special kinds o geome ic s uc u es on a gi en smoo h mani old
Ma e con olled by PDEs.
A possible way o cons uc hese s uc u es is o ind a Lie g oup Gac ing on M
in such a way ha he de ining PDE is in a ian unde he ac ion o G. In gene al he
dimension o ou p oblem will be educed, and i will su ice o cons uc a solu ion
on a submani old ans e sal o he o bi s o Gon M, since his solu ion will be
anspo ed by he ac ion o G o he es o M. The simples scena io happens
when Gac s ansi i ely on M, and hen he PDE u ns in o an algeb aic equa ion.
Howe e , i Gac s wi h cohomogenei y one on M, ou ini ial PDE will be educed o an
o dina y di e en ial equa ion. These symme y educ ion me hods, and pa icula ly
cohomogenei y one me hods, ha e been ex emely use ul and success ul on he sea ch
o geome ic s uc u es on Riemannian mani olds.
35
42 3 Homogeneous hype su aces in symme ic spaces
o Mde ined a ound p. No ice ha , in his case, Mis cu a u e adap ed i and only
i Dis in a ian by he shape ope a o o M, whe e Dis he maximal subbundle o
he angen bundle o M ha is in a ian unde J, see [12].
Case (1) in Theo em 3.2.4 induces he ac ion o H=SpnSp1on HPngi en by
he iso opy ac ion o HPn=Spn+1/Spn×Sp1. This ac ion has geodesic sphe es
S4n−3=SpnSp1/Spn−1Sp1as p incipal o bi s, and a poin and a o ally geodesic
HPn−1as singula o bi s. Mo eo e , any p incipal o bi H·phas g= 2 dis inc
p incipal cu a u es. The eigenspaces o he shape ope a o associa ed wi h ξpa e
Jξpand TpM⊖Jξp, which ha e dimensions 3 and 4n−4, espec i ely.
The case (2) induces he ac ion o H=Spk+1 ×Spn−kon HPn, which has a o ally
geodesic HPk+1 and a o ally geodesic HPn−kas singula o bi s, and ubes a ound
any o hese o ally geodesic submani olds as p incipal o bi s. The p incipal o bi s a e
equal o Spk+1 ×Spn−k/(Spk×Spn−k−1×Sp1), up o a quo ien by a ini e subg oup.
In his case any p incipal o bi H·phas g= 3 dis inc p incipal cu a u es. Le
q= (q1, q2) = π−1(p)∈S4n+3 ⊂Hn+1 and assume ha qi∈Vi⊂Hn+1 is no ze o,
whe e V1and V2a e he qua e nionic subspaces in a ian unde he ac ions o Spk+1
and Spn−k, espec i ely. Then, he eigenspaces o he shape ope a o associa ed wi h
ξpa e π∗q(Vi), o i∈ {1,2}, and Jξp. The dimensions o π∗q(V1) and π∗q(V2) a e 4k
and 4(n−k−1), espec i ely.
Finally, case (3) induces he ac ion o H=Un+1 on HPn, which has as singula
o bi s a o ally geodesic submani old isome ic o CPnand a minimal homogeneous
space ha is equal o Un+1/(Un−1×SU2), up o a quo ien by a ini e subg oup.
E e y p incipal o bi is a ube a ound any o hese singula o bi s and i is equal
o he homogeneous space Un+1/(Un−1×S(U1×U1)), up o a quo ien by a ini e
subg oup. In his case any p incipal o bi H·phas g= 4 dis inc p incipal cu a u es.
Le q= (q1, q2) = π−1(p)∈S4n+3 ⊂Hn+1 and assume ha qi∈Wi⊂Hn+1 is no
ze o, whe e W1 he subspace in a ian unde mul iplica ion by he imagina y uni i
induced by Hand W2is i s o hogonal complemen in Hn+1. Obse e ha Hlea es
Wiin a ian o e e y i∈ {1,2}. The eigenspaces o Sξpa e RJ1ξp, span{J2ξp, J3ξp}
and π∗qWi o i∈ {1,2}. The dimension o π∗qWiis 2(n−1).
The classi ica ion p oblem in OP2was sol ed by Iwa a [100].
Theo em 3.2.6. A hype su ace in OP2is homogeneous i and only i i is:
(1) a geodesic sphe e, o
(2) a ube a ound a o ally geodesic HP2in OP2.
Fi s ly, geodesic sphe es o OP2can be ega ded as p incipal o bi s o he iso opy
ac ion o Spin9on OP2and, as homogeneous spaces, hey a e isomo phic o S15 =
Spin9/Spin7and ha e g= 2 dis inc p incipal cu a u es. The singula o bi s o his
ac ion a e a ixed poin and i s cu locus, namely, a o ally geodesic OP1=S8=
Spin9/Spin8.
The second hype su ace can be ega ded as a p incipal o bi o he ac ion o
Sp3Sp1on OP2. Any such p incipal o bi is isomo phic o he homogeneous space
Sp3Sp1/(Sp1×Sp1×Sp1) and has g= 4 dis inc p incipal cu a u es (see [141]).

3.2 Cohomogenei y one ac ions: he compac case 43
The singula o bi s o his ac ion a e a o ally geodesic HP2and a minimal S11 =
Sp3Sp1/Sp2Sp1.
In Table 3.1, we lis he homogeneous hype su aces o symme ic spaces o com-
pac ype and ank one, and hei ocal se s, up o a quo ien by a ini e subg oup.
These we e compu ed in [118]. We deno e by ρn, µnand νn he s anda d ep esen-
a ions o SOnon Rn,SUn(o Un) on Cnand Spnon C2n, espec i ely. Mo eo e ,
we deno e by Ad he adjoin ep esen a ion, by λ3 he 14-dimensional i educible
ep esen a ion o Sp3o qua e nionic ype, by λ4 he 26-dimensional i educible ep-
esen a ion o F4, by ∆+
10 he hal -spin ep esen a ion o Spin10 (see [193, Chap e 5]),
and we w i e −θ o omi he 1-dimensional i al ep esen a ion.
In 1998, in a monumen al wo k o he heo y o homogeneous hype su aces,
Koll oss [113] classi ied homogeneous hype su aces in i educible symme ic spaces
o compac ype. Be o e s a ing his esul , we will ecall some well-known ac s abou
He mann ac ions.
Le Gbe a compac semisimple Lie g oup equipped wi h a bi-in a ian me ic and
Hand Kbe closed symme ic subg oups o G. This means ha Hand Ka e ixed
poin se s o in olu i e au omo phisms o G. The e o e, (G,H) and (G,K) a e compac
symme ic pai s. Unde hese condi ions, we say ha a He mann ac ion is he ac ion
o H×Kon Ggi en by
(h, k)·g=hgk−1, h ∈H, k ∈K, g ∈G.
Clea ly, his ac ion induces a na u al ac ion o Hon he compac symme ic space
G/K. In addi ion o ha , i u ns ou ha he slice ep esen a ion o H×Kon Gis he
same as he slice ep esen a ion o he ac ion o Hon G/K. Hence, he ac ion H×K
on Ghas cohomogenei y one i and only i he ac ion o Hon G/Khas cohomogenei y
one. Mo eo e , all He mann ac ions a e hype pola and hei o ally geodesic o bi s
a e e lec i e, see [143].
The main idea o he wo k by Koll oss is he ollowing. Le M=G/Kbe an
i educible symme ic space o compac ype. We s a om he op o he la ice o
subalgeb as o g, he Lie algeb a o G, downwa ds un il we ge a subalgeb a co e-
sponding o a connec ed subg oup o Gac ing wi h cohomogenei y one. Once we ha e
achie ed such a subg oup we s op going down h ough ha b anch and we choose a
di e en b anch o he la ice. This wo ks because i his p ope ly con ained in a sub-
algeb a h′whose co esponding connec ed subg oup H′o Gac s wi h cohomogenei y
k, hen Hac s wi h cohomogenei y g ea e o equal han k. Fu he mo e, i H⊂H′
and H′ac s wi h cohomogenei y one, we ha e ha ei he Hac s wi h he same o bi s
as H′o Hac s wi h la ge cohomogenei y.
Rema k 3.2.7.As an example o he me hods used by Koll oss, we p o ide a p oo o
he classi ica ion o homogeneous hype su aces in OP2, which was o iginally ob ained
by Iwa a ia opological a gumen s.
Le OP2=G/K, whe e G=F4and K=Spin9. Le Hbe a subg oup o Gac ing
wi h cohomogenei y one. By [150, P oposi ion 3, p. 45], he complexi ica ion o a
maximal subalgeb a ho a simple compac Lie algeb a gis maximal in g⊗C. Then h,
44 3 Homogeneous hype su aces in symme ic spaces
Symme ic space Rep esen a ion P incipal O bi Singula O bi s n g (m1, m2)
S1×Sℓ−1, ℓ ≥3ρℓ−1Sℓ−2{∗} ℓ−1 1 ℓ−2
Sk+1 ×Sℓ−k−1ρk+1 +ρℓ−k−1Sk×Sℓ−k−2Sk,Sℓ−k−2ℓ−1 2 (k, ℓ −k−2)
SU3/SO3Sym2ρ3−θSO3RP24 3 1
SU3Ad SU3/T2CP27 3 2
SU6/Sp3Λ2ν3−θSp3/(Sp1)3HP213 3 4
E6/F4λ4F4/Spin8OP225 3 8
Sp2Ad SO5/(SO2×SO2)SO5/(SO2×SO3) 9 4 (2,2)
SO10/U5(Λ2µ5)RSU5/(SU2×SU2)SU5/(SU2×SU3),SU5/SO519 4 (4,5)
E6/Spin10U1(µ⊗C∆+
10)RSpin10/Spin6Spin10/SU5,Spin10/Spin731 4 (9,6)
SOk+2/(SO2×SOk), k ≥3ρ2⊗RρkS1×SOk/SOk−2S1×Sk−1,SOk/SOk−22k−1 4 (1, k −2)
SUk+2/S(U2×Uk), k ≥2 (µ2⊗Cµk)RS2×Uk/Uk−2S2×S2k−1,Uk/Uk−24k−1 4 (2,2k−3)
Spk+2/(Sp2×Spk), k ≥3ν2⊗HνkS4×Spk/Spk−2S4×S4k−1,Spk/Spk−28k−1 4 (4,4k−5)
Sp4/(Sp2×Sp2)ν2⊗Hν2S7×S7S4×S7,Sp215 4 (1,3)
G2/SO4(Sym3µ2⊗Cµ2)RSO4SO4/SO27 6 (1,1)
G2Ad G2/T2G2/U213 6 (2,2)
Table 3.1: Homogeneous hype su aces in Sn.
Symme ic space Rep esen a ion P incipal O bi Singula O bi s n g Mul iplici ies
CP1×CPℓ, ℓ ≥2µ1+µℓS2ℓ−1{∗},CPℓ−1ℓ2 (1,2(ℓ−1))
CPk+1 ×CPℓ−k,1≤k≤ℓ−2µk+1 +µℓ−kUk+1 ×Uℓ−k−1/(Uk×Uℓ−k−2×U1)CPk,CPℓ−k−1ℓ3 (1,2(ℓ−1−k),2k)
SOℓ+3/(SO2×SOℓ+1), ℓ ≥2ρ2⊗RρℓSOℓ+1/SOℓ−1RPℓ,SOℓ+1/(SOℓ−1SO2)ℓ3 (1, ℓ −1, ℓ −1)
SUℓ+3/S(U2×Uℓ+1), ℓ ≥2 (µ2⊗Cµℓ+1)RCP1×Uℓ+1/(Uℓ−1U1)CP1×CPℓ,Uℓ+1/(Uℓ−1U1) 2ℓ+ 1 5 (1,2,2,2(ℓ−1),2(ℓ−1))
SO10/U5(Λ2µ5)RSU5/S(U2×U2)SU5/S(U2×U3),SU5/SO5U19 5 (1,4,4,4,4)
E6/Spin10U1(µ⊗C∆+
10)RSpin10/Spin6U1Spin10/U5,Spin10/Spin7U115 5 (1,6,6,8,8)
Table 3.2: Homogeneous hype su aces in CPn.
Symme ic space Rep esen a ion P incipal O bi Singula O bi s gMul iplici ies
HP1×HPnν1+νnS4n−3{∗},HPn−12 (3,4n−6)
HPk+1 ×HPn−k,1≤k≤n−2νk+1 +νn−kSpk+1 ×Spn−k/(Spk×Spn−k−1×Sp1)HPk,HPn−k−13 (3,4k, 4(n−k−1))
SUn+3/S(U2×Un+1) (µ2⊗Cµn+1)RUn+1/(Un−1×S(U1×U1)) CPn,Un+1/(Un−1×SU2) 4 (1,2,2(n−1))
Table 3.3: Homogeneous hype su aces in HPn.
G oup ac ing Rep esen a ion P incipal O bi Singula O bi s gMul iplici ies
Spin9∆4S15 {∗},OP12 (7,8)
Sp3Sp1λ3⊗µ2Sp3Sp1/(Sp1×Sp1×Sp1)HP2,S11 4 (4,4,3,4)
Table 3.4: Homogeneous hype su aces in OP2.
3.2 Cohomogenei y one ac ions: he compac case 45
he Lie algeb a o H, is con ained in one o he ollowing maximal subalgeb as (see [74,
Table 12 and Table 39]) o 4:
so9,sp3⊕sp1,g1
2⊕su8
2,su156
2,
whe e we indica e wi h a supe sc ip he Dynkin index o he complexi ied subalgeb a,
see Sec ion
§
7.4. No ice ha he subg oups o Gco esponding o so9and sp3⊕
sp1ac wi h cohomogenei y one, since hey co espond o he iso opy ac ion o
OP2, and o he ac ion whose p incipal o bi s a e ubes a ound a o ally geodesic
submani old o OP2isome ic o HP2, espec i ely. Thus, i we p o ed ha he
subg oups co esponding o g2⊕su8
2and su156
2ac wi h cohomogenei y la ge han
one, we would be done.
The case o su156
2 ollows easily. Obse e ha , in gene al, i His a subg oup ac ing
on M, we ha e
dim(H)≥dim H·p= (dim H·p−dim M) + dim M= dim M−cohom(H·p).
Hence, o he case co esponding o a subalgeb a su156
2we canno ha e cohomogenei y
one ac ions by dimensional easons.
Obse e ha he p e ious a gumen does no apply o he case o g2⊕su8
2. A
possible way o ackle his case is o s udy he slice ep esen a ion o he ac ion o he
H-ac ion on OP2a he base poin o∈OP2, which coincides wi h he cohomogenei y
o he ac ion o Hon OP2. No ice ha he iso opy o his ac ion a he base poin
is equal o H∩K. Le us assume ha h=g2⊕su8
2. We know ha he ac o o
hisomo phic o g2is maximally con ained in a subalgeb a isomo phic o spin7in
k∼
=spin9. Howe e , g2⊕su8
2canno be con ained in spin9, since o he wise i would
no be a maximal semisimple subalgeb a o 4. Hence, h∩kis ei he equal o g2
o o g2⊕u1. Howe e , hese subalgeb as ha e dimension 14 and 15, espec i ely,
and we know ha he co esponding connec ed Lie subg oups canno ac ansi i ely
on sphe es o dimension dim νo(H·o)−1 = dim OP2−dim H/G2−1 = 12 and
dim νo(H/(G2U1))−1 = 13, espec i ely. This p o es ha he subg oup co esponding
o g2⊕su8
2ac s wi h cohomogenei y la ge han one.
The classi ica ion heo em ob ained by Koll oss can be s a ed as ollows.
Theo em 3.2.8. Le M=G/Kbe an i educible symme ic space o compac ype.
A cohomogenei y one ac ion on Mis locally o bi equi alen o one o he ollowing
ac ions:
(1) a He mann ac ion o cohomogenei y one (see Table 3.5), o
(2) he ac ion o {(g, ¯g) : g∈SU3}on SU3, o
(3) an ac ion induced by he iso opy ep esen a ion o a symme ic space o ank
wo, o
(4) one o he se en excep ions co esponding o he ac ion o H×Kon G, o he
ac ion o Hon G/K, whe e (H,K,G)is a iple appea ing in Table 3.6.
46 3 Homogeneous hype su aces in symme ic spaces
H G K
SOn+1 SUn+1 S(Un×U1)
S(U2×U2n−2)SU2nSpn
S(U3×U2n−3)SU2nSpn
S(Up+q×U1)SUp+q+1 S(Up×Uq+1)
SOp+qSOp+q+1 SOp×SOq+1
Spn×Sp1Spn+1 Un+1
Spp+q×Sp1Spp+q+1 Spp×Spq+1
SO2×SO2n−2SO2nUn
SO3×SO2n−3SO2nUn
SU6·SU2E6F4
SO10 ·SO2E6F4
Sp3·Sp1F4Spin9
Table 3.5: Ac ions in i em (1) o Theo em 3.2.8.
H G2G2U3Spin9Sp1SpnSU3SU3
K SO3×SO4G2G2SO2×SO14 SO2×SO4n−2SO4SU3
G SO7SO7SO7SO16 SO4nG2G2
Table 3.6: Ac ions in i em (4) o Theo em 3.2.8.
3.3 A dig ession: he gene alized K¨ahle angle
In his sec ion we ecall he no ion o gene alized K¨ahle angle o a ec o wi h espec
o a subspace o a Cli o d module in oduced in [60]. This no ion cons i u ed a
gene aliza ion o he no ion o K¨ahle angle o a ec o o a eal subspace in Cn,
see [36] o [13]. This concep will be o g ea ele ance o he s udy o cohomogenei y
one ac ions on hype bolic spaces, see Sec ion
§
3.4. We will s a by ecalling his olde
no ion be o e in oducing he gene alized K¨ahle angle.
Le us endow Cnwi h he inne p oduc gi en by he eal pa o i s s anda d
He mi ian inne p oduc . Le V⊂Cnbe a eal subspace. Fu he mo e, le us deno e
by πV:Cn→V he o hogonal p ojec ion on o V, and by J:Cn→Cn he linea
map gi en by he mul iplica ion by he imagina y uni i∈C.
The K¨ahle angle o a non-ze o ec o ∈Vwi h espec o Vis gi en by
∡(J , V ), he angle be ween J and V. Equi alen ly, his is he alue φ∈[0, π/2]
such ha
||πVJ ||2= cos2(φ)|| ||2.
The eal subspace V⊂Cnhas cons an K¨ahle angle φ∈[0, π/2] i
∡(J , V ) = φ o e e y ∈V {0}.
No ice ha he e a e wo ex eme cases. On he one hand, V⊂Cnhas cons an
K¨ahle angle 0 i and only i i is complex, i.e. i is in a ian unde J. On he o he
hand, V⊂Cnhas cons an K¨ahle angle π/2 i and only i i is o ally eal, i.e.
3.3 A dig ession: he gene alized K¨ahle angle 47
JV ⊂V⊥=Cn⊖V. Thus, he K¨ahle angle p o ides a way o measu e how a eal
subspace o Cn ails o be complex.
Example 3.3.1.Le {e1, e2}be he canonical basis o C2and conside he eal subspace
V= span{e1,cos(φ)Je1+ sin(φ)e2} o some φ∈[0, π/2]. We claim ha Vhas
cons an K¨ahle angle φ. Le =ae1+b(cos(φ)Je1+ sin(φ)e2), whe e a2+b2= 1.
Now,
J =aJe1+b(−cos(φ)e1+ sin(φ)Je2),
and hen πVJ =−bcos(φ)e1+acos(φ)(cos(φ)Je1+ sin(φ)e2). Consequen ly,
||πVJ ||2= cos2(φ)(b2+a2) = cos2(φ),
p o ing ou claim.
Indeed, e e y eal subspace o cons an K¨ahle angle φ∈[0, π/2] is equal o he
di ec sum o copies o he subspace Vin Example 3.3.1, up o some ans o ma ion
T∈Un. E en mo e, e e y eal subspace o Cncan be ac o ized as a di ec sum o
eal subspaces wi h cons an K¨ahle angle as i was shown in [61]. He e we include
an al e na i e p oo o his ac .
Theo em 3.3.2. Le V⊂Cnbe a eal subspace. Then, Vadmi s an unique o hog-
onal decomposi ion gi en by
V=M
φ∈Φ
Vφ,
whe e
i) Vφhas cons an K¨ahle angle φ∈[0, π/2], and
ii) CVφ⊥CVψ, o e e y dis inc φ, ψ ∈Φ.
P oo . Le k= dim Vand deno e by S 2(V) = SOk/SOk−2 he S ie el mani old o o -
hono mal 2- ames in V, and conside :S 2(V)→R, which maps a 2-o hono mal
ame (u, )∈S 2(V) o ⟨Ju, ⟩. By compac ness, he e exis s some (u1, 1)∈
S 2(V) such ha ⟨Ju1, 1⟩= cos(φ1) is a maximum o . Obse e ha cos(φ1)=0
i and only i Vis o ally eal. Assume Vis no o ally eal. Then, since u1and 1
a e pe pendicula , he e exis s some unique uni ec o w1∈V⊖spanR{u1, Ju1, J 1}
ha sa is ies 1= cos(φ1)Ju1+ sin(φ1)w1. An analogous compu a ion as in Exam-
ple 3.3.1 shows ha he subspace V1
φ1:= spanR{u1, 1}has cons an K¨ahle angle
φ1∈[0, π/2].
In wha ollows, we will p o e ha V=V1
φ1⊕(V⊖V1
φ1) is a C-o hogonal di ec
sum. Le w∈V⊖Vφ1, and conside he map gw: [0,2π]→R, gi en by gw(θ) :=
⟨Ju1,cos(θ) 1+ sin(θ)w⟩ o each θ∈[0,2π]. Now, since a ains a maximum a
(u1, 1), gwa ains a maximum a θ= 0. Hence, ⟨Ju1, w⟩= 0 and using an analogous
a gumen one p o es ha ⟨J 1, w⟩= 0. Thus, V=V1
φ1⊕(V⊖V1
φ1) is a C-o hogonal
di ec sum.
Now we p oceed induc i ely, and since Vis ini e dimensional, we end up ac o -
izing Vas an o hogonal sum V=L
i=1 Ls
j=1 Vj
φi, whe e Vj
φihas cons an K¨ahle

48 3 Homogeneous hype su aces in symme ic spaces
angle φi. Mo eo e , CVj
φi⊥CVj′
φi′i i=i′o j=j′. Consequen ly, i one se s
Vφi:= Ls
j=1 Vj
φi, o e e y i∈ {1, . . . , }, we ha e he desi ed o hogonal decompo-
si ion o V. Finally, no ice ha his decomposi ion is unique by cons uc ion.
P oposi ion 3.3.3. Le V⊂Cnbe a eal subspace. Then, he ollowing s a emen s
a e equi alen :
i) Vhas cons an K¨ahle angle.
ii) The e is some Lie subg oup H⊂Unac ing ansi i ely on he uni sphe e o V.
P oo . Le us assume ha Vhas cons an K¨ahle angle equal o π/2. Then i has
dimension k≤n. Now, i we conside CV, he complex span o V, he e is some
subg oup isomo phic o Ukin Unwhich ac s ansi i ely on he uni sphe e o CV.
Howe e , now he e is a subg oup Okin Ukac ing ansi i ely on he uni sphe e
o V.
Now assume ha Vhas cons an K¨ahle angle φ < π/2. By he p oo o The-
o em 3.3.2, Vhas e en dimension k≤2n. Hence, he map P:V→Vgi en by
P( ) = 1
cos(φ)πVJ de ines a complex s uc u e on V, which p ese es ⟨·,·⟩. Thus,
U(V) = {g∈GL(V) : gP =Pg, g g= IdV}is a g oup isomo phic o Uk. Any elemen
g∈U(V) can be ex ended o A∈Unby de ining A( 1+J 2) = g 1+Jg 2and
Aw =w, o e e y 1, 2∈Vand w∈Cn⊖CV. This shows ha U(V) can be e-
ga ded as a subg oup o he no malize o Vin Un. The g oup U(V) ac s ansi i ely
on he uni sphe e o V. This p o es ha i) implies ii).
Now le us p o e ha ii) implies i). Le , w ∈V⊂Cnbe wo uni ec o s.
Then, he e is T∈H⊂Unsuch ha T =wand TV ⊂V. Le us assume ha
has K¨ahle angle φwi h espec o V. Then,
cos2(φ) = ⟨πVJ , πVJ ⟩=⟨T πVJ , T πVJ ⟩=⟨πVTJ , πVTJ ⟩
=⟨πVJT , πVJT ⟩=⟨πVJw, πVJw⟩,
whe e we ha e used ha T∈Unand ha i p ese es V. Then, he K¨ahle angle
o wwi h espec o Vis also φ∈[0, π/2], and V⊂Cnhas cons an K¨ahle angle
φ∈[0, π/2].
Le Mk,n be he moduli space o non-ze o eal subspaces o eal dimension kin
Cnwi h cons an K¨ahle angle. Then, Mk,n is desc ibed in he ollowing able.
Mk,n 1≤k≤n n < k ≤2n
kodd {π/2} ∅
ke en [0, π/2] {0}
Table 3.7: Se s o possible K¨ahle angles o k-dimensional eal subspaces o Cn.
In he ollowing lines we will ecall he no ion o gene alized K¨ahle angle in o-
duced in [60]. Le be a Cli o d module o e Cl(z, q) and J:z→End( ) he e-
s ic ion o zo he unde lying Cli o d algeb a ep esen a ion, see Subsec ion
§
1.4.1
3.4 Cohomogenei y one ac ions on hype bolic spaces 49
o some basic ac s abou Cli o d modules o ep esen a ions. We equip zwi h he
inne p oduc induced by pola iza ion o −q, and ex end i o an inne p oduc ⟨·,·⟩
on n= ⊕z, so ha and za e pe pendicula , and JZis an o hogonal map o each
uni Z∈z. Then, ncan be na u ally endowed wi h a gene alized Heisenbe g algeb a
s uc u e as de ined in Subsec ion
§
1.4.2.
Le wbe a subspace o . We deno e by w⊥= ⊖w he o hogonal complemen
o win . Fo each Z∈zand ξ∈w, we w i e JZξ=PZξ+FZξ, whe e PZξand FZξ
deno e he o hogonal p ojec ions o JZξon o wand w⊥, espec i ely.
Then, ξ∈w,ξ= 0, is said o ha e K¨ahle angle φ∈[0, π/2] wi h espec o
he elemen Z∈z(o wi h espec o JZ) and he subspace w⊂ i ⟨PZξ, PZξ⟩=
cos2(φ)⟨Z, Z⟩⟨ξ, ξ⟩.
The ollowing heo em, c . [60, Theo em 3.1], is undamen al o unde s and he
de ini ion o gene alized K¨ahle angle which will be in oduced inmedia ely a e .
Theo em 3.3.4. Le wbe some ec o subspace o and le ξ∈wbe a non-ze o
ec o . Then he e exis s an o hono mal basis {Z1, . . . , Zm}o zand a uniquely
de ined m- uple (φ1, . . . , φm)such ha :
(i) φiis he K¨ahle angle o ξwi h espec o JZi, o each i= 1, . . . , m.
(ii) ⟨PZiξ, PZjξ⟩=⟨FZiξ, FZjξ⟩= 0 whene e i=j.
(iii) 0≤φ1≤φ2≤ ··· ≤ φm≤π/2.
(i ) φ1is minimal and φmis maximal among he K¨ahle angles o ξwi h espec o
all he elemen s o z.
Thus, he gene alized K¨ahle angle o ξwi h espec o wis he m- uple (φ1, . . . , φm)
sa is ying p ope ies (i)-(i ) o Theo em 3.3.4.
Rema k 3.3.5.Obse e ha he K¨ahle angles φ1, . . . , φmdepend, no only on he
subspace wo , bu also on he ec o ξ∈wand he basis {Z1, . . . , Zm}.
A subspace wo has cons an gene alized K¨ahle angle (φ1, . . . , φm) i he m-
uple (φ1, . . . , φm) is independen o he uni ec o ξ∈w.
3.4 Cohomogenei y one ac ions on hype bolic
spaces
In his sec ion we explain he gene al heo y o cohomogenei y one ac ions on hype -
bolic spaces de eloped by Be nd and Tama u. Recall ha a quick in oduc ion on
hype bolic spaces ( ank one symme ic spaces o non-compac ype) was de eloped
in Subsec ion
§
1.4.3.
We can dis inguish h ee di e en classes o cohomogenei y one ac ions on sym-
me ic spaces o non-compac ype and ank one, up o o bi equi alence. I was
shown in [13] ha any such ac ion has a mos one singula o bi . The ex insic ge-
ome y o hese singula o bi s and hei ubes, which a e homogeneous hype su aces,
was s udied in [60].
50 3 Homogeneous hype su aces in symme ic spaces
Ac ions wi h no singula o bi
Be nd and Tama u [26] classi ied ac ions wi hou singula o bi s on hype bolic
spaces FHn. They p o ed ha he e a e exac ly wo such ac ions up o o bi equi a-
lence. The Iwasawa decomposi ion G=KAN associa ed wi h FHnplays an impo an
ole in he desc ip ion o hese ac ions (we e e o Sec ion
§
1.4 o no a ion).
(i) The ac ion o Non FHnhas cohomogenei y one. The o bi s o his ac ion a e
mu ually cong uen ho osphe es ha o m a egula Riemannian olia ion on
FHn, called he ho osphe e olia ion.
(ii) Le Sbe he connec ed Lie subg oup o AN wi h Lie algeb a s=a⊕w⊕z, whe e
wis a ec o subspace o gαo codimension one. The ac ion o Son FHnhas
cohomogenei y one and i s o bi s o m a egula Riemannian olia ion on FHn,
called he sol able olia ion. Di e en choices o wlead o conjuga e ac ions.
Ac ions wi h a o ally geodesic singula o bi
Be nd and B ¨uck [13] classi ied cohomogenei y one ac ions on FHnwi h a o-
ally geodesic singula o bi F. In o de o do so i su ices o classi y hose o ally
geodesic submani olds Fwhose ubes a e homogeneous hype su aces. Since o ally
geodesic submani olds in symme ic spaces o ank one a e classi ied (see Figu e 8.1),
i su ices o compu e he associa ed slice ep esen a ion in o de o check when he
cohomogenei y o he co esponding ac ion is one.
The ubes a ound a o ally geodesic submani old Fo FHna e homogeneous i
and only i Fis one o he o ally geodesic submani olds lis ed below:
(i) F=R:F∈ {poin ,RH1, . . . , RHn−1};
(ii) F=C:F∈ {poin ,CH1, . . . , CHn,RHn+1};
(iii) F=H:F∈ {poin ,HH1, . . . , HHn,CHn+1};
(i ) F=O:F∈ {poin ,OH1,HH2}.
Ac ions wi h a non- o ally geodesic singula o bi
Be nd and Tama u [28] ga e a cons uc ion me hod o all cohomogenei y one
ac ions wi h a non- o ally geodesic singula o bi in hype bolic spaces. Such ac ions
only appea i F=R. We ecall ha K0ac s on he oo space gαby he adjoin
ep esen a ion, and hence, i Vis a eal subspace o gα,N0
K0(V) will deno e he
connec ed componen o he iden i y o he no malize o Vin K0.
Theo em 3.4.1. Le g=k⊕a⊕nbe an Iwasawa decomposi ion o he Lie algeb a
o he isome y g oup o he hype bolic space M=FHn,F∈ {C,H,O}.
3.4 Cohomogenei y one ac ions on hype bolic spaces 51
(i) Le Vbe a non-ze o ec o subspace o gαsuch ha N0
K0(V)ac s ansi i ely on
he uni sphe e o V. Deno e by gα⊖V he o hogonal complemen o Vin gα.
Then he connec ed subg oup o Gwi h Lie algeb a
Nk0(V)⊕a⊕(gα⊖V)⊕g2α
ac s on Mwi h cohomogenei y one, and he o bi h ough ois singula , p o ided
ha dim V≥2. Fu he mo e, e e y cohomogenei y one ac ion on Mwi h a non-
o ally geodesic singula o bi can be ob ained in his way up o o bi equi alence.
(ii) Le Vand V′be ec o subspaces o gαas in i em (i), and assume ha he co -
esponding cohomogenei y one ac ions ha e non- o ally geodesic singula o bi s.
Then, hese ac ions a e o bi equi alen i and only i he e exis s k∈K0such
ha Ad(k)V=V′.
In he case o eal hype bolic spaces, we ha e al eady seen ha homogeneous
hype su aces ha e cons an p incipal cu a u es and hese a e classi ied, see Sec-
ion
§
2.2. Thus, we only ha e o deal wi h hype bolic spaces o e C,Ho O.
Be nd and Tama u in [28] classi ied homogeneous hype su aces in CHn. Thei
esul can be s a ed as ollows:
Theo em 3.4.2 (Homogeneous hype su aces in complex hype bolic spaces).A ho-
mogeneous hype su ace in CHnis cong uen o:
(i) a geodesic sphe e, o
(ii) a ube a ound a o ally geodesic CHkin CHn,k∈ {1, . . . , n −1}, o
(iii) a ube a ound a o ally geodesic RHnin CHn, o
(i ) a ho osphe e, o
( ) a uled homogeneous minimal Lohnhe hype su ace W2n−1
π/2, o one o i s equidis-
an hype su aces, o
( i) a ube a ound a uled homogeneous minimal Be nd –B ¨uck submani old W2n−k
φ,
o k∈ {2, . . . , n −1}, wi h φ∈(0, π/2], whe e kis e en i φ=π/2.
The i s h ee examples a ise by conside ing he dual ac ion o he co esponding
cohomogenei y one ac ion in he complex p ojec i e space CPn, see Sec ion
§
3.2 o
he classi ica ion and [115] o a de ailed exposi ion on dual ac ions. These a e o bi s
o he ac ions o he ollowing subg oups o SU1,n:
S(U1×Un),S(U1,k ×Un−k),SO0
1,n,
whe e k∈ {1, . . . , n −1}, espec i ely.
The homogeneous hype su aces in (i ) co espond o he o bi s o he ac ion o
he g oup N, which a e isome ic o gene alized Heisenbe g g oups, and olia e CHn.
Mo eo e , he o bi s o his ac ion a e p incipal and cong uen o each o he .

Chap e 4
Homogeneous hype su aces in HHn
The aim o his chap e is o p esen he classi ica ion o homogeneous hype su aces
in qua e nionic hype bolic spaces, and hus, o conclude he classi ica ion o homoge-
neous hype su aces in ank one symme ic spaces. This had been an open p oblem
o mo e han wen y yea s. As a by-p oduc o ou s udy, we cons uc o he
i s ime uncoun ably many inhomogeneous isopa ame ic amilies o hype su aces
wi h cons an p incipal cu a u es in Riemannian mani olds. These esul s ha e been
published in [63].
The i s main esul o his chap e can be s a ed in e ms o qua e nionic alge-
b a. We deno e by H he eal di ision algeb a o he qua e nions, endowed wi h i s
s anda d complex s uc u es i,jand k. Le Hnbe a igh qua e nionic ec o space
o dimension n. The compac symplec ic g oup Spnis he g oup o qua e nionic ma-
ices (ac ing on he le on Hn) ha p ese e he s anda d qua e nionic bilinea o m
Pn
i=1 ¯ iwi, whe e , w ∈Hn, and ba deno es conjuga ion. This bilinea o m na u-
ally induces an inne p oduc in Hn ha makes i isome ic wi h he Euclidean space
R4n. By Jwe will deno e he qua e nionic s uc u e o Hn, ha is, he subspace o
eal endomo phisms o Hngene a ed by he igh mul iplica ions by i, j and k, which
can he e o e be seen as he Lie algeb a o Sp1.
We also conside he Lie g oup Sp1Spn=Sp1×Spn/Z2, which ac s on Hnas (q, A)·
=A q−1. This is an impo an g oup in di e en ial geome y, as i a ises in Be ge ’s
holonomy lis , ha is, he lis o Lie g oups which can be ealized as he holonomy
o i educible, simply connec ed and non-locally symme ic Riemannian mani olds.
Thus, a Riemannian mani old is called qua e nionic K¨ahle i i has dimension 4n,
is no Ricci- la , and i s holonomy is isomo phic o a subg oup o Sp1Spn,n≥2.
The simples examples o symme ic, qua e nionic K¨ahle spaces a e he qua e nionic
p ojec i e spaces, and hei non-compac duals, he qua e nionic hype bolic spaces.
In any case, unde s anding algeb aic p ope ies linked o holonomy g oups is a i s
undamen al s ep owa ds he s udy o mo e geome ic ques ions, such as hose ela ed
o cu a u e (e.g. he celeb a ed LeB un-Salamon conjec u e [121]) o submani olds
(e.g. he heo y o calib a ions [40]). Simila ly, he p oblem o submani old geome y
ha we add ess in his chap e elies on a linea algeb aic p oblem ha we desc ibe
below.
We say ha a eal subspace Vo Hnis p o ohomogeneous i he e exis s a con-
nec ed Lie subg oup o Sp1Spn ha ac s ansi i ely on he uni sphe e o V. A
p o ohomogeneous subspace o Hnhas cons an qua e nionic K¨ahle angle, which is a
pa icula ins ance o he no ion o subspace wi h cons an gene alized Kahle angle
o a Cli o d module, in oduced in Sec ion
§
3.3. As his concep is cen al in ou
59
60 4 Homogeneous hype su aces in HHn
s udy, we shall ecall i now in he cu en qua e nionic se ing. Le πVdeno e he
o hogonal p ojec ion on o a ec o subspace V, and de ine
PJ=πV◦J, whe e J∈J.
We say ha Vhas cons an qua e nionic K¨ahle angle (φ1, φ2, φ3), wi h φ1≤φ2≤
φ3, i o any ∈V he symme ic bilinea o m
L :J×J→R, L (J, J′) = ⟨PJ , PJ′ ⟩,
has eigen alues cos2(φi)⟨ , ⟩,i∈ {1,2,3}. We poin ou he e he ac ha he bilin-
ea o ms L , ∈V, desc ibed abo e do no necessa ily diagonalize simul aneously
(al hough we can p o e a p io i ha hey do so o p o ohomogeneous subspaces o
dimension g ea e han o equal o 5, see Co olla y 4.3.2, and by classi ica ion esul s
o dimension di e en om 3).
The i s main esul o his chap e is o classi y p o ohomogeneous subspaces o
Hn, up o cong uence by elemen s in Sp1Spn. We p esen he e he moduli space o
such subspaces o dimension kin Hnby exhibi ing hei possible qua e nionic K¨ahle
angles. In Theo em A, and in wha ollows, ⊔deno es disjoin union.
Theo em A. The moduli space Mk,n o non-ze o p o ohomogeneous subspaces o
dimension kin Hn, up o cong uence in Sp1Spn, is desc ibed in he ollowing able:
Mk,n k≤n n < k ≤4n
3
4n
3< k ≤2n k > 2n
k≡0 (mod 4) (R+
4 R−
4)⊔(R−
4×Z2)S{(0, φ, φ)}φ∈[0,π
2]{(0,0,0)}
k≡2 (mod 4) {(φ, π
2,π
2)}φ∈[0,π
2]{(0,π
2,π
2)} {(0,π
2,π
2)} ∅
k= 3 odd {(π
2,π
2,π
2)} ∅ ∅ ∅
k= 3 (R+
3 R−
3)⊔(R−
3×Z2)∅ {(φ, φ, π
2)}φ∈{0,π
3}{(0,0,π
2)}
whe e Λ = {(φ1, φ2, φ3)∈[0, π/2]3:φ1≤φ2≤φ3}, and
R+
3={(φ, φ, π/2) ∈Λ : φ∈[0, π/2]},
R−
3={(φ, φ, π/2) ∈Λ : φ∈[π/3, π/2)},
R+
4={(φ1, φ2, φ3)∈Λ : cos(φ1) + cos(φ2)−cos(φ3)≤1},
R−
4={(φ1, φ2, φ3)∈Λ : cos(φ1) + cos(φ2) + cos(φ3)≤1, φ3=π/2},
S={(φ1, φ2, φ3)∈Λ : cos(φ1) + cos(φ2) + εcos(φ3) = 1, o ε=±1}.
This classi ica ion includes ypical examples such as o ally eal subspaces (p e-
cisely hose wi h qua e nionic K¨ahle angle (π/2, π/2, π/2)), o ally complex sub-
spaces (wi h qua e nionic K¨ahle angle (0, π/2, π/2)), qua e nionic subspaces (wi h
qua e nionic K¨ahle angle (0,0,0)), subspaces o cons an K¨ahle angle φ∈(0, π/2)
inside a o ally complex ec o subspace (wi h qua e nionic K¨ahle angle equal o
(φ, π/2, π/2)), complexi ica ions o subspaces o cons an K¨ahle angle φ∈(0, π/2)
in a o ally complex subspace (wi h qua e nionic K¨ahle angle (0, φ, φ)), and J ,
∈Hn, = 0 (wi h qua e nionic K¨ahle angle (0,0, π/2)). Howe e , he e a e some
o he non-classical examples. Some o hem we e in oduced in [60], bu he e a e
some o he s, which a e basically p esen ed and classi ied in Sec ion
§
4.4. A basis
61
o hese subspaces can be calcula ed explici ly, bu o R±
3and R±
4i s exp ession is
a he long. See P oposi ion 4.4.3 o R±
3and P oposi ions 4.4.10 and 4.5.1 o R±
4 o
ge u he de ails. Fu he mo e, he e a e non-cong uen subspaces o Hnwi h he
same K¨ahle angles. These co espond p ecisely o he in e sec ions R+
3∩R−
3=R−
3
and R+
4∩R−
4=R−
4.
We poin ou he e h ee main ools ha ha e been essen ial o ob ain his classi-
ica ion. Fi s we use he classical gene aliza ion o he hai y ball heo em ega ding
he possible ank o con inuous dis ibu ions on sphe es [166] in o de o educe he
classi ica ion p oblem o eal subspaces o Hnwi h cons an qua e nionic K¨ahle angle
o subspaces o dimensions 3 and mul iples o 4 (Sec ion
§
4.2). Secondly, we p o ide
a Lie heo e ic a gumen elying on esul s by Bo el [38] and Mon gome y and Samel-
son [136] on g oups ac ing e ec i ely and ansi i ely on sphe es, o p o e ha , o
subspaces o dimension g ea e o equal han 5, he maps L ha a e used o de-
ine qua e nionic K¨ahle angle diagonalize simul aneously (Co olla y 4.3.2). In hi d
place, using he p e ious esul s, we can show ha a p o ohomogeneous subspace o
dimension 4lis he sum o p o ohomogeneous subspaces o dimension 4 wi h he same
qua e nionic K¨ahle angle (Subsec ion
§
4.3.2). All his educes he classi ica ion o
p o ohomogeneous subspaces o dimensions 3 and 4. A his s age, we ac ually ob ain
he mo e gene al classi ica ion o eal subspaces o dimensions 3 and 4 wi h cons an
qua e nionic K¨ahle angle. This is a (ha d) p oblem o linea algeb a ha is sol ed
in Sec ion
§
4.4.
The i s consequence o Theo em A is he classi ica ion o cohomogenei y one
ac ions on qua e nionic hype bolic spaces HHn+1 up o o bi equi alence. In ac ,
Be nd and Tama u explained in [28] a me hod o ob ain his classi ica ion. This
me hod was explained in Sec ion
§
3.4 o he gene al se ing o hype bolic spaces,
bu we shall pa icula ize i now in he cu en qua e nionic se ing. Conside he
symme ic pai (G,K)=(Sp1,n+1,Sp1×Spn+1) ep esen ing he symme ic space
HHn+1. We deno e by g=k⊕p he co esponding Ca an decomposi ion, and le abe
a maximal abelian subspace o p, which is one-dimensional because HHn+1 is o ank
one. Le g=g−2α⊕g−α⊕g0⊕gα⊕g2αbe he es ic ed oo space decomposi ion o
gwi h espec o a. Then, gαis isomo phic o a qua e nionic ec o space Hnendowed
wi h he s anda d qua e nionic bilinea o m, and K0∼
=Sp1×Spn, he connec ed Lie
subg oup o Gwhose Lie algeb a is k0=g0∩k=Nk(a), no malizes gαand ac s on gαin
he canonical way. The classi ica ion o cohomogenei y one ac ions on HHn+1 can be
ob ained i we de e mine he p o ohomogeneous subspaces Vo gα∼
=Hn. I Vis such
a p o ohomogeneous subspace, we de ine he Lie subalgeb a sV=a⊕(gα⊖V)⊕g2α
o g, and deno e by SV he connec ed Lie subg oup o Gwi h Lie algeb a sV. We
ecall ha ⊖deno es o hogonal complemen . Then N0
K0(SV)SV=N0
K0(V)SVac s on
HHn+1 wi h cohomogenei y one, whe e N0
K0(·) deno es he connec ed componen o
he iden i y o he no malize in K0. Knowing all such subspaces Vup o cong uence
by an elemen o Sp1Spnde e mines all cohomogenei y one ac ions on HHn+1 up o
o bi equi alence.
Roughly wen y yea s a e Be nd and B ¨uck [13] announced he i s examples
o cohomogenei y one ac ions using his p ocedu e, we ob ain he ull classi ica ion o
62 4 Homogeneous hype su aces in HHn
cohomogenei y one ac ions on qua e nionic hype bolic spaces up o o bi equi alence
as a consequence o Theo em A. Toge he wi h he esul s by Be nd and Tama u [28],
his inishes he classi ica ion o cohomogenei y one ac ions on non-compac symme ic
spaces o ank one:
Theo em B. The moduli space o cohomogenei y one ac ions on HHn+1 up o o bi
equi alence is gi en by he disjoin union
{N,K,SU1,n+1}⊔
4n
G
k=1 Mk,n.
The ac ions e e enced he e a e:
(1) N: he ac ion ha p oduces a ho osphe e olia ion.
(2) K: he ac ion ha p oduces a amily o geodesic sphe es cen e ed a a poin .
(3) SU1,n+1: he ac ion ha p oduces a amily o ubes a ound a o ally geodesic
CHn+1.
(4) Mk,n: he cohomogenei y one ac ions o he connec ed Lie subg oups o Sp1,n+1
wi h Lie algeb as Nk0(V)⊕a⊕(gα⊖V)⊕g2α, whe e Vis a p o ohomogeneous
subspace o dimension ko gα∼
=Hn.
We no e ha , in his classi ica ion, he ac ion o Sp1,ℓ ×Spn+1−ℓ⊂Sp1,n+1, which
gi es ubes a ound a o ally geodesic lowe dimensional qua e nionic hype bolic space
HHℓ,ℓ∈ {1, . . . , n}, in HHn+1, is included in i em (4), whe e in his case Vis a
qua e nionic subspace o gα∼
=Hn(hence, o qua e nionic K¨ahle angle (0,0,0)) o
eal dimension k= 4(n−ℓ+ 1). Mo eo e , i we ake Va line in gα(i.e. k= 1), hen
N0
K0(V) is i ial and we eco e he ac ion ha gi es ise o he so-called sol able
olia ion [26].
In ou s udy o p o ohomogeneous subspaces o Hnwe ha e also encoun e ed
non-cong uen pai s o subspaces wi h he same cons an qua e nionic K¨ahle angles.
Mo eo e , we p o e in Sec ion
§
4.5 ha an H-o hogonal di ec sum o subspaces o
dimension 4 wi h he same cons an qua e nionic K¨ahle angle is p o ohomogeneous
i and only i any wo ac o s a e cong uen unde an elemen o Spn. Howe e , e en
i ha di ec sum is no p o ohomogeneous, i has cons an qua e nionic K¨ahle angle
in some cases. Thus, i we ake Va non-p o ohomogeneous subspace wi h cons an
qua e nionic K¨ahle angle as abo e, and deno e by SV he connec ed subg oup o
Gwhose Lie algeb a is sV=a⊕(gα⊖V)⊕g2α, hen: (1) since Vhas cons an
qua e nionic K¨ahle angle, ubes a ound SV·oa e isopa ame ic and ha e cons an
p incipal cu a u es by [60, Theo em 4.5], and (2) hese ubes a e no homogeneous
by [28, Theo em 4.1]. Hence, we ha e he ollowing ema kable consequence:
Theo em C. The e exis uncoun ably many inhomogeneous isopa ame ic amilies
o hype su aces wi h cons an p incipal cu a u es in HHn+1 wi h n≥7, up o con-
g uence.
4.1 Qua e nionic K¨ahle angle 63
We ecall ha he only examples o inhomogeneous isopa ame ic amilies o hy-
pe su aces wi h cons an p incipal cu a u es known so a in any i educible Rie-
mannian symme ic space a e he celeb a ed examples in sphe es by Fe us, Ka che
and M¨unzne [78] and a single example ound in he Cayley hype bolic plane [60].
Thus, his is he i s ime an uncoun able collec ion o such examples is p oduced in
some symme ic space.
This chap e is o ganized as ollows. The undamen al concep o qua e nionic
K¨ahle angle is ecalled in Sec ion
§
4.1 oge he wi h some impo an no a ion ha
will be used h oughou his chap e . In Sec ion
§
4.2 we use a gene aliza ion o he
hai y ball heo em o ule ou se e al possibili ies o qua e nionic K¨ahle angles.
Then, in Subsec ion
§
4.3.1 we p o e a simul aneous diagonaliza ion esul o sub-
spaces o cons an qua e nionic K¨ahle angle. This is used in Subsec ion
§
4.3.2 o
p o e a ac o iza ion heo em o p o ohomogeneous subspaces o dimension mul iple
o 4. Al oge he , his educes ou s udy o dimensions 3 (Subsec ion
§
4.4.1) and 4
(Subsec ion
§
4.4.2). The exis ence o inhomogeneous isopa ame ic hype su aces
wi h cons an p incipal cu a u es in qua e nionic hype bolic spaces (Theo em C) is
es ablished in Sec ion
§
4.5. We inally p o e Theo ems A and B in Sec ion
§
4.6.
4.1 Qua e nionic K¨ahle angle
We s a his sec ion by in oducing he main known esul s conce ning he con-
cep o qua e nionic K¨ahle angle, which we ecall in his sec ion. Also, we will p esen
some p ope ies and summa ize all he examples o subspaces wi h cons an qua e -
nionic K¨ahle angle known up o he p esen . The main e e ences o hese no ions
and esul s a e [13], [28], and [60].
The me ic and he qua e nionic K¨ahle s uc u e on HHn+1 induce a posi i e
de ini e inne p oduc ⟨·,·⟩ on gαand a qua e nionic s uc u e Jon gα, espec i ely,
such ha gαis isomo phic o Hnas a ( igh ) qua e nionic Euclidean space. He e, by a
qua e nionic s uc u e Jwe unde s and a 3-dimensional ec o subspace o EndR(Hn),
he space o eal endomo phisms o Hn∼
=R4n, admi ing a basis {J1, J2, J3}o
o hogonal ans o ma ions o Hn∼
=R4nsuch ha J2
i=−Id and JiJi+1 =Ji+2 =
−Ji+1Ji, o each i∈ {1,2,3}(indices modulo 3). Such a basis is called a canonical
basis o he qua e nionic s uc u e J. Some imes i is help ul o ega d Jas endowed
wi h a posi i e de ini e inne p oduc ha makes i isome ic o he Euclidean 3-space
R3, and such ha he elemen s o J ha a e o hogonal complex s uc u es o Hn
cons i u e he uni sphe e S2⊂Jwi h espec o such inne p oduc . Th oughou his
chap e , i is a ec o in Hnand Vis a eal subspace o Hn(i.e. a ec o subspace
o he eal ec o space R4nwi h he unde lying eal ec o space s uc u e o Hn),
we deno e by H =R ⊕J and by HV=V+JV he qua e nionic spans o ∈Hn
and o V⊂Hn, espec i ely; some imes we also w i e (Im H) o e e o J .
Theo em 3.4.1, due o Be nd and Tama u, shows he c ucial ole played by eal
subspaces Vo gα∼
=Hnand hei beha io wi h espec o K0in he classi ica ion
p oblem o cohomogenei y one ac ions on HHn+1. No e ha he e ec i iza ion o K0
on gα∼
=Hnis he Lie g oup Sp1Spn= (Sp1×Spn))/{±(1,Id)}, which ac s in he

64 4 Homogeneous hype su aces in HHn
s anda d way: (q, A)· =A q−1, whe e q∈Sp1and A∈Spn. Thus, in his subsec ion
we ga he some impo an e minology and use ul ac s o s udy eal subspaces o a
qua e nionic Euclidean space, up o cong uence by elemen s o Sp1Spn.
Fi s ly, mo i a ed by Theo em 3.4.1, we will say ha a eal subspace V⊂Hnis
p o ohomogeneous i he e is a connec ed subg oup o Sp1Spn ha ac s ansi i ely on
he uni sphe e o V. Equi alen ly, Vis p o ohomogeneous i he connec ed Lie g oup
N0
Sp1Spn(V) ac s ansi i ely on he uni sphe e o V. No e ha p o ohomogeneous
subspaces Vo gα∼
=Hna e p ecisely hose inducing cohomogenei y one ac ions on
HHn+1 ia he cons uc ion in Theo em 3.4.1(i). We also say ha wo eal subspaces
Vand Wo Hna e equi alen i he e exis s an elemen T∈Sp1Spnsuch ha TV =
W. Obse e ha , by Theo em 3.4.1(ii), Vand Wa e equi alen p o ohomogeneous
subspaces o gα∼
=Hni and only i hey induce o bi equi alen cohomogenei y one
ac ions on HHn+1.
Le us now ecall a use ul desc ip ion o he ac ion o Sp1Spnon Hn. Le us conside
{X1, . . . , Xn}and {Y1, . . . , Yn} wo H-o hono mal bases o Hn, and le {J1, J2, J3}
and {J′
1, J′
2, J′
3}be wo canonical bases o he qua e nionic s uc u e o Hn. Then,
he e exis s a unique T∈Sp1Spnsuch ha T(Xi) = Yiand TJj=J′
jT o all
i∈ {1, . . . , n}and all j∈ {1,2,3}. Con e sely, any R-linea endomo phism o Hn
ha maps H-o hono mal bases o Hn o H-o hono mal bases o Hnand in e wines
canonical bases o he qua e nionic s uc u e o Hnin he abo e desc ibed ashion lies
in Sp1Spn.
Le Vbe a eal ec o subspace o he qua e nionic Euclidean space Hn. The
K¨ahle angle o a non-ze o ec o ∈Vwi h espec o a non-ze o J∈Jand Vis
de ined o be he angle be ween J and V. Equi alen ly, i is he alue φ∈[0, π/2]
such ha ⟨PJ , PJ ⟩= cos2(φ)⟨ , ⟩, whe e PJ:= πVJand we deno e by πV he
o hogonal p ojec ion on o V.
The ollowing lemma was essen ially p o ed by Be nd and B ¨uck [13, Lemma 3].
We s a e i in a somewha di e en o m ollowing Theo em 3.3.4, whe e i was s a ed
in he mo e gene al con ex o subspaces o Cli o d modules.
Lemma 4.1.1. Le Vbe a eal subspace o Hnand le ∈Vbe a non-ze o ec o .
Then he e exis s a canonical basis {J1, J2, J3}o Jand a uniquely de ined iple
(φ1, φ2, φ3), such ha :
(i) φiis he K¨ahle angle o wi h espec o Ji o each i∈ {1,2,3},
(ii) ⟨Pi , Pj ⟩= 0 o e e y i=j, whe e Pi=πVJi.
(iii) φ1≤φ2≤φ3.
(i ) φ1is minimal and φ3is maximal among he K¨ahle angles o wi h espec o
all non-ze o elemen s o J.
Indeed, {J1, J2, J3}is a basis o Jwi h espec o which he symme ic bilinea o m
L :J×J→R, L (J, J′) := ⟨PJ , PJ′ ⟩,
has a diagonal ma ix exp ession wi h eigen alues cos2(φi)⟨ , ⟩,i∈ {1,2,3}.
4.1 Qua e nionic K¨ahle angle 65
The p e ious lemma allows us o in oduce he ollowing de ini ion [13]. I Vis a
eal subspace o Hn, he qua e nionic K¨ahle angle o a non-ze o ec o ∈Vwi h
espec o Vis he iple (φ1, φ2, φ3) gi en in Lemma 4.1.1. Some imes we will also
say ha ∈Vhas qua e nionic K¨ahle angle (φ1, φ2, φ3) wi h espec o Vand o
he canonical basis {J1, J2, J3}o J, in o de o speci y ha he basis {J1, J2, J3}
is unde he condi ions o Lemma 4.1.1. A linea subspace Vo Hnis said o ha e
cons an qua e nionic K¨ahle angle Φ(V) = (φ1, φ2, φ3) i he iple (φ1, φ2, φ3) is
independen o he non-ze o (o by linea i y, uni ) ec o ∈V. In his chap e ,
whene e we use he no a ion Φ(V) we will implici ly assume ha Vhas cons an
qua e nionic K¨ahle angle.
Rema k 4.1.2.No e ha he Ji∈Jde ined in Lemma 4.1.1 may depend on ∈V.
This is ue, e en in he case ha Vhas cons an qua e nionic K¨ahle angle. Fo
example V= Im H⊂Hhas cons an qua e nionic K¨ahle angle Φ(V) = (0,0, π/2),
bu he basis {J1, J2, J3}o Lemma 4.1.1 canno be chosen independen ly o ∈V.
Howe e , we will p o e ha , unde ce ain hypo heses (see Co olla y 4.3.2 o P opo-
si ion 4.4.10), he Jican be chosen independen ly o ∈V. This is one o he c ucial
esul s in his chap e .
The ollowing esul is known (see [13, p. 229]), bu we ind i ins uc i e o
include a p oo .
Lemma 4.1.3. Le V⊂Hnbe a p o ohomogeneous subspace. Then, Vhas cons an
qua e nionic K¨ahle angle.
P oo . Le ∈Vbe a uni ec o o qua e nionic K¨ahle angle (φ1, φ2, φ3) wi h
espec o Vand a canonical basis {J1, J2, J3}o J. Thus, ⟨Pi , Pj ⟩= cos2(φi)δij, o
i, j ∈ {1,2,3}, whe e δij s ands o K onecke del a. Le w∈Vbe a uni ec o . Since
Vis p o ohomogeneous, he e exis s T∈Sp1Spn ha lea es Vin a ian and sa is ies
T =w. By he desc ip ion o he ac ion o Sp1Spnon Hn, he e exis s a canonical
basis {J′
1, J′
2, J′
3}o Jsuch ha TJi=J′
iT, o i∈ {1,2,3}. Fu he mo e, since T
lea es Vin a ian , we ha e ha TπV=πVT. Hence, TPi=P′
iT o i∈ {1,2,3},
whe e P′
i=PJ′
i=πVJ′
i. Finally,
⟨P′
iw, P′
jw⟩=⟨P′
iT , P ′
jT ⟩=⟨T Pi , TPj ⟩=⟨Pi , Pj ⟩= cos2(φi)δij .
Since wis a bi a y, by he las claim o Lemma 4.1.1 we ge Φ(V) = (φ1, φ2, φ3).
We now in oduce a ma ix map ha will be e y use ul in wha ollows. Le V
be a eal subspace o Hno dimension k, and le {J1, J2, J3}be a canonical basis o
J. Then, we de ine he K¨ahle angle map o Vwi h espec o {J1, J2, J3}as he map
Ω ha sends each uni ec o ∈Sk−1⊂V o he symme ic ma ix Ω( ) o o de 3
whose (i, j)-en y is gi en by
Ω( )ij := ⟨Pi , Pj ⟩=L (Ji, Jj),(4.1)
whe e Pi=PJi,i∈ {1,2,3}. A s aigh o wa d bu impo an obse a ion is ha
Vhas cons an qua e nionic K¨ahle angle i and only i he ma ices Ω( ) ha e he
66 4 Homogeneous hype su aces in HHn
same eigen alues coun ed wi h mul iplici ies, o any ∈Sk−1. In o he wo ds,
Φ(V) = (φ1, φ2, φ3) i and only i he eigen alues o Ω( ) a e cos2(φi), i∈ {1,2,3},
o all uni ∈V. This isospec ali y p ope y o he K¨ahle angle map will play a
c ucial ole in his chap e .
Known examples o subspaces wi h cons an qua e nionic K¨ahle angle
We conclude his sec ion by s a ing some known pa ial classi ica ions and ex-
amples o subspaces Vwi h cons an qua e nionic K¨ahle angle in a qua e nionic
Euclidean space Hn.
In [28], Be nd and Tama u lis ed some iples ha can a ise as cons an qua e -
nionic K¨ahle angles Φ(V) o non-ze o eal subspaces Vo Hn, and s a ed he clas-
si ica ion o such pa icula ypes o subspaces. All he subspaces in his lis a e
p o ohomogeneous [13, 28]. Such iples a e he ollowing:
(1) Φ(V) = (π/2, π/2, π/2). These a e p ecisely he o ally eal subspaces o Hn.
Recall ha a linea subspace V⊂Hnis o ally eal i JV ⊂Hn⊖V o e e y
J∈J. In his case dimRV∈ {1,2, . . . , n}.
(2) Φ(V) = (0, π/2, π/2). These a e he o ally complex subspaces, ha is, he
subspaces Vo Hnsuch ha J1V⊂Vand JV ⊂Hn⊖V o some complex
s uc u e J1∈Jand all J∈Jpe pendicula o J1. In his case dimRV∈
{2,4, . . . , 2n}.
(3) Φ(V) = (0,0, π/2). These subspaces a e he 3-dimensional subspaces o he
o m J = (Im H) o some non-ze o ∈Hn.
(4) Φ(V) = (0,0,0). These a e he qua e nionic subspaces, ha is, he subspaces
V⊂Hnsuch ha JV ⊂V o e e y J∈J. Hence, dimRV∈ {4,8, . . . , 4n}.
(5) Φ(V) = (φ, π/2, π/2), φ∈(0, π/2). Le Wbe a o ally complex subspace o
Hn, wi h J1W⊂W o some complex s uc u e J1∈J. Then, a subspace V
o Hnsa is ies Φ(V) = (φ, π/2, π/2) i and only i Vis a subspace o some W
as be o e wi h cons an K¨ahle angle φ∈(0, π/2) as a subspace o he complex
ec o space (W, J1). Thus dimRV∈ {2,4, . . . , 2[n/2]}.
(5) Φ(V) = (0, φ, φ). Le Wbe a o ally complex subspace o Hnsuch ha J2W⊂
W o some complex s uc u e J2∈J, and le ˜
Vbe a eal subspace o (W, J2)
wi h cons an K¨ahle angle φ∈(0, π/2). Then, Vis a subspace o Hnwi h
Φ(V) = (0, φ, φ) i and only i i is he complexi ica ion V=J1˜
V⊕˜
Vo some
˜
V⊂Was be o e wi h espec o some complex s uc u e J1∈Jo hogonal o
J2. In his case dimRV∈ {4,8, . . . , 4[n/2]}.
We also ecall, as obse ed in [28, pp. 3434-3435], ha :
(i) o each ℓ∈ {1, . . . , n} he e exis s, up o equi alence, exac ly one eal subspace
Vo Hnwi h dimRVequal o ℓ, 2ℓo 4ℓ, o each o he ypes (1), (2) o (4)
abo e, espec i ely;
4.2 Hai y ball me hod 67
(ii) he e exis s only one subspace Vo Hno ype (3), up o equi alence; and
(iii) o each ℓ∈ {1, . . . , [n/2]}and each φ∈(0, π/2) he e exis s exac ly one sub-
space Vo Hnwi h dimRV= 2ℓo ype (5), and exac ly one subspace Vo Hn
wi h dimRV= 4ℓo ype (6), up o equi alence.
Be nd and Tama u conjec u ed in [28] ha hese we e all he possible subspaces
wi h cons an qua e nionic K¨ahle angle, bu in [60] new examples o subspaces Vo
dimension 4 such ha Φ(V) = (φ1, φ2, φ3) whe e cos(φ1) + cos(φ2)<1 + cos(φ3)
we e gi en. These a e cons uc ed as ollows. Le 0 < φ1≤φ2≤φ3≤π/2 wi h
cos(φ1) + cos(φ2)<1 + cos(φ3), and conside a 4-dimensional o ally eal subspace
o Hnand a basis o uni ec o s {e0, e1, e2, e3}o i , whe e ⟨e0, ei⟩= 0, o i∈ {1,2,3},
and
⟨ei, ei+1⟩=cos(φi+2)−cos(φi) cos(φi+1)
sin(φi) sin(φi+1), i ∈ {1,2,3}.
Fo he sake o simplici y le us de ine φ0= 0 and J0= Id. No ice ha ⟨Jjek, el⟩= 0
o j∈ {1,2,3}and k,l∈ {0,1,2,3}, because spanR{e0, e1, e2, e3}is a o ally eal
subspace o Hn. Then we can de ine
ξk= cos(φk)Jke0+ sin(φk)Jkek, k ∈ {0,1,2,3}.
(No e ha ξ0=e0.) We conside he subspace Vspanned by hese ou ec o s, o
which {ξ0, ξ1, ξ2, ξ3}is an o hono mal basis. Then, Φ(V)=(φ1, φ2, φ3). I was also
obse ed in [60] ha one can ake se e al copies o hese 4-dimensional subspaces o
cons uc subspaces Vo Hno dimension mul iple o 4 wi h Φ(V)=(φ1, φ2, φ3),
whe e cos(φ1)+cos(φ2)<1+cos(φ3). This ac will be p o ed ca e ully in Sec ion
§
4.5
o a b oade amily o examples ha we will p o ide.
A his poin , we ind in e es ing o ema k ha , unlike he six ypes o examples
known o Be nd and Tama u in [28], and as we will see in P oposi ion 4.4.14, we
can p o e ha , o any posi i e in ege kmul iple o 4, he e a e iples (φ1, φ2, φ3)
o which he e a e non-equi alen subspaces Vo Hnwi h Φ(V) = (φ1, φ2, φ3) and
dimRV=k.
4.2 Hai y ball me hod
In his sec ion we use a opological a gumen o educe he classi ica ion p oblem o
subspaces Vwi h cons an qua e nionic K¨ahle angle in Hn o he s udy o subspaces
wi h dimensions 3 and mul iples o 4. The idea is o cons uc a dis ibu ion on he
uni sphe e o he subspace Vo Hn, and hen use a gene aliza ion o he hai y ball
heo em o exclude se e al cases.
Le Vbe a eal subspace o Hno eal dimension kwi h cons an qua e nionic
K¨ahle angle Φ(V) = (φ1, φ2, φ3). Le Sk−1deno e he uni sphe e o V. Fo each
∈Sk−1and J∈Jwe ha e ⟨PJ , ⟩= 0 and PJ ∈V, and hus PJ ∈T Sk−1. Fo
each ∈Sk−1conside he subspace o T Sk−1gi en by
∆ ={PJ :J∈J}.
74 4 Homogeneous hype su aces in HHn
4.4.1 Subspaces o dimension h ee
In his subsec ion we classi y 3-dimensional eal subspaces o Hnwi h cons an
qua e nionic K¨ahle angle.
P oposi ion 4.4.3. Le V⊂Hnbe a eal subspace o dimension 3. Then, Vhas
cons an qua e nionic K¨ahle angle i and only i Φ(V) = (φ, φ, π/2),φ∈[0, π/2],
and o any uni e0∈V, he e is a canonical basis {J1, J2, J3}o Jsuch ha
{e0,cos(φ)J1e0+ sin(φ)J1e1,cos(φ)J2e0+ sin(φ)J2e2}(4.3)
is an o hono mal basis o V, whe e, i φ= 0,e1, e2a e uni ec o s sa is ying
e1, e2∈Hn⊖He0,e2∈Hn⊖(Im H)e1, and ei he ⟨e1, e2⟩= cos(φ)/(cos(φ)−1) wi h
φ∈[π/3, π/2], o ⟨e1, e2⟩= cos(φ)/(cos(φ) + 1) wi h φ∈(0, π/2].
P oo . By P oposi ion 4.2.2, we ha e ha φ1=φ2=φ∈[0, π/2] and φ3=π/2. Le
us assume ha Vis spanned by he basis desc ibed in Lemma 4.4.1 wi h k= 3. I
φ= 0 o φ=π/2 he claim ollows om he classi ica ion o subspaces wi h cons an
qua e nionic K¨ahle angle (0,0, π/2) o (π/2, π/2, π/2); see Sec ion
§
4.1.
Thus, le us assume φ∈(0, π/2). Then, o each l∈ {1,2}and unde s anding he
subsc ip l+ 1 ∈ {1,2}modulo 2,
Ω( ¯
Ple0)ij =⟨Pi¯
Ple0, Pj¯
Ple0⟩
=⟨Ji¯
Ple0, e0⟩⟨Jj¯
Ple0, e0⟩+
2
X
=1⟨Ji¯
Ple0,¯
P e0⟩⟨Jj¯
Ple0,¯
P e0⟩,
=⟨¯
Ple0, Pie0⟩⟨¯
Ple0, Pje0⟩+⟨Ji¯
Ple0,¯
Pl+1e0⟩⟨Jj¯
Ple0,¯
Pl+1e0⟩,
whe e in he second equali y we ha e calcula ed he o hogonal p ojec ion o ec o s
on o Vby using he o hono mal basis {e0,¯
P1e0,¯
P2e0}o V. Hence, o l∈ {1,2},
using Lemma 4.4.1 we ha e
Ω( ¯
Ple0)ll = cos2(φ) + ⟨el, Jl+1el+1⟩2sin4(φ),
Ω( ¯
Ple0)l+1,l =⟨e1, J1e2⟩⟨e1, J2e2⟩sin4(φ),
Ω( ¯
Ple0)l+1,l+1 =⟨el+1, Jlel⟩2sin4(φ),
Ω( ¯
Ple0)13 =⟨e2, J2e1⟩sin2(φ)(cos2(φ) + ⟨e1, e2⟩sin2(φ)),
Ω( ¯
Ple0)23 =−⟨e1, J1e2⟩sin2(φ)(cos2(φ) + ⟨e1, e2⟩sin2(φ)),
Ω( ¯
Ple0)33 = (cos2(φ) + ⟨e1, e2⟩sin2(φ))2.
(4.4)
Now, since Ω( ¯
Ple0) is symme ic wi h eigen alues cos2(φ) (o mul iplici y 2) and 0,
by he min-max heo em, one ob ains
0≤Ω( ¯
Ple0)ll ≤cos2(φ), l ∈ {1,2}.
This implies ⟨e1, J2e2⟩=⟨e2, J1e1⟩= 0, which oge he wi h Rema k 4.4.2 yields
e2∈Hn⊖(Im H)e1. Taking again in o accoun he spec um o Ω( ¯
P1e0), we ha e he
ollowing ela ion o i s ace,
2 cos(φ)2= (Ω( ¯
P1e0)) = (cos2(φ) + ⟨e1, e2⟩sin2(φ))2+ cos2(φ).

4.4.1 Subspaces o dimension h ee 75
F om his and he ac ha e1and e2a e uni ec o s, we deduce ha ei he ⟨e1, e2⟩=
cos(φ)/(cos(φ)−1) whe e φ∈[π/3, π/2) o ⟨e1, e2⟩= cos(φ)/(1 + cos(φ)) whe e
φ∈(0, π/2). This p o es he necessi y o he s a emen .
Fo he con e se we ake an a bi a y uni ec o ∈Vwhich we w i e as
=x0e0+x1cos(φ)J1e0+ sin(φ)J1e1+x2cos(φ)J2e0+ sin(φ)J2e2.
Then, i ε∈ {±1}is such ha ⟨e1, e2⟩= cos(φ)/(1 + εcos(φ)), we ha e
Ω( ) = cos2(φ)
x2
0+x2
1x1x2−εx0x2
x1x2x2
0+x2
2εx0x1
−εx0x2εx0x1x2
1+x2
2
.
Since is a uni ec o , x2
0+x2
1+x2
2= 1, and i is now easy o see ha Ω( ) has a
double eigen alue cos2(φ), and a simple eigen alue 0.
Rema k 4.4.4.We will deno e by Vφ
+and Vφ
−any eal subspace o Hncons uc ed
as in P oposi ion 4.4.3, depending on whe he ⟨e1, e2⟩= cos(φ)/(cos(φ) + 1) o
φ∈(0, π/2], o ⟨e1, e2⟩= cos(φ)/(cos(φ)−1) o φ∈[π/3, π/2], espec i ely. No e
ha he subspaces Vφ
±can be cons uc ed as subspaces o any Hnwi h n≥3. One
can easily check ha he only one ha i s in o an H2is Vπ/3
−(bu i canno i in o
H).
P oposi ion 4.4.5. Le Vbe a subspace o Hnwi h cons an qua e nionic K¨ahle
angle and dimension 3. Then Vis p o ohomogeneous.
P oo . We know om P oposi ion 4.4.3 ha Φ(V) = (φ, φ, π/2). We can assume ha
φ∈(0, π/2) since, o he wise, Vis known o be p o ohomogeneous (see Sec ion
§
4.1).
Le e0∈Vbe an a bi a y uni ec o . By Lemma 4.1.1 he e is a canonical
basis {J1, J2, J3}o Jsuch ha e0has K¨ahle angle φwi h espec o J1and J2, and
K¨ahle angle π/2 wi h espec o J3. In iew o Lemma 4.4.1 and P oposi ion 4.4.3,
le us conside he uni ec o s ei∈Hn⊖He0,i∈ {1,2}, gi en by
ei:= −(Ji¯
Pie0+ cos(φ)e0)/sin(φ), i ∈ {1,2},(4.5)
whe e ¯
Pi:= πVJi/cos(φ). On he one hand, by (4.5) we ha e
⟨e1, e2⟩=1
sin2(φ)⟨J1¯
P1e0+ cos(φ)e0, J2¯
P2e0+ cos(φ)e0⟩
=1
sin2(φ)⟨J1¯
P1e0, J2¯
P2e0⟩−cos2(φ).
(4.6)
On he o he hand, again by P oposi ion 4.4.3, ⟨e1, e2⟩can ake wo possible alues.
We will i s see ha , gi en V,⟨e1, e2⟩is independen o e0.
Le S2deno e he uni sphe e o V. We de ine Θ: S2→Rby Θ(e0) = ⟨e1, e2⟩.
We claim ha Θ is well de ined. Le {J′
1, J′
2, J3}be ano he canonical basis o Jsuch
ha e0has K¨ahle angle φwi h espec o J′
i,i∈ {1,2}, and le e′
i:= −(J′
i¯
P′
ie0+
76 4 Homogeneous hype su aces in HHn
cos(φ)e0)/sin(φ) whe e ¯
P′
i:= πVJ′
i/cos(φ) o i∈ {1,2}. Then, he e is θ∈[0,2π)
such ha J′
i= cos(θ)Ji+ (−1)i+1 sin(θ)Ji+1 o i∈ {1,2}and subsc ip s modulo 2.
Thus,
J′
i¯
P′
i= (cos(θ)Ji+ (−1)i+1 sin(θ)Ji+1)(cos(θ)¯
Pi+ (−1)i+1 sin(θ)¯
Pi+1)
= cos2(θ)Ji¯
Pi+ sin2(θ)Ji+1 ¯
Pi+1 + (−1)i+1 cos(θ) sin(θ)(J1¯
P2+J2¯
P1).(4.7)
Consequen ly, using Equa ion (4.6) wice, and hen (4.7), we ge , a e some calcula-
ions,
⟨e1, e2⟩−⟨e′
1, e′
2⟩=1
sin2(φ)⟨J1¯
P1e0, J2¯
P2e0⟩−⟨J′
1¯
P′
1e0, J′
2¯
P′
2e0⟩= 0,
which implies ha Θ is well-de ined.
Now no e ha he assignmen e0∈S27→ span{J1, J2} ∈ G2(J), whe e G2(J) is he
G assmannian o 2-planes o J∼
=R3, is con inuous due o he con inuous dependence
o he quad a ic o m J∈J7→ L (J, J) = ⟨PJ , PJ ⟩ ∈ Ron . Hence, he map Θ is
also con inuous. Bu , as men ioned jus a e (4.6), Θ(S2) has a mos wo elemen s.
The e o e, Θ is cons an on S2.
Finally, we p o e ha Vis p o ohomogeneous. Le e0,e′
0be a bi a y uni ec o s
in V. Le {J1, J2, J3},{J′
1, J′
2, J′
3}be canonical bases o J, and e1, e2, e′
1, e′
2be uni
ec o s in Vsuch ha bo h (4.3), and (4.3) wi h e′
iins ead o eiand J′
iins ead
o Ji, a e o hono mal bases o V. Bo h se s o ec o s {e0, e1, e2}and {e′
0, e′
1, e′
2}
span a o ally eal subspace o Hn, and since Θ is cons an , ⟨ei, ej⟩=⟨e′
i, e′
j⟩ o all
i, j ∈ {0,1,2}. I hen ollows ha he e exis s an elemen T∈Sp1Spnsuch ha
Tei=e′
i o each i∈ {0,1,2}, and TJj=J′
jT o each j∈ {1,2,3}. Thus, by (4.5)
we ge T¯
Pie0=¯
P′
ie′
0 o i∈ {0,1,2}, whe e ¯
P0=¯
P′
0= Id. The e o e, Tis an elemen
o Sp1Spnsuch ha TV =Vand Te0=e′
0. Since e0, e′
0a e a bi a y, his p o es
ha Vis p o ohomogeneous.
Finally we show ha he wo ypes o subspaces Vφ
+and Vφ
−in oduced in Re-
ma k 4.4.4 a e indeed inequi alen o φ=π/2. Recall ha Vφ
+is de ined o all
φ∈(0, π/2], bu Vφ
−only o φ∈[π/3, π/2].
P oposi ion 4.4.6. Le φ∈[π/3, π/2]. Then he e exis s T∈Sp1Spnsuch ha
TV φ
+=Vφ
−i and only i φ=π/2.
P oo . I φ=π/2, hen Vπ/2
+and Vπ/2
−a e o ally eal, he e o e equi alen . Le us
assume ha φ=π/2 and ha he e is T∈Sp1Spnsuch ha TV φ
+=Vφ
−. By applying
an elemen o Sp1Spni necessa y, we can assume ha he e is a uni ec o e0∈
Vφ
+∩Vφ
−and ha e0has qua e nionic K¨ahle angle (φ, φ, π/2) wi h espec o bo h
Vφ
+and Vφ
−and a common canonical basis {J1, J2, J3}o J. Then, by Lemma 4.4.1
and P oposi ion 4.4.3, Vφ
±is spanned by he basis {e0,¯
P±
1e0,¯
P±
2e0}, whe e ¯
P±
i:=
πVφ
±Ji/cos(φ), i∈ {1,2}. Mo eo e ,
¯
P±
ie0= cos(φ)Jie0+ sin(φ)Jie±
i,wi h ⟨e±
1, e±
2⟩=cos(φ)
cos(φ)±1,
4.4.2 Subspaces o dimension ou 77
and e±
i∈Hn⊖He0,i∈ {1,2}.
By P oposi ion 4.4.5, we can assume ha Te0=e0. Le J′=TJ3T−1∈J.
Since J3e0∈Hn⊖Vφ
+, we ha e J′e0=J′Te0=TJ3e0∈Hn⊖Vφ
−. This implies
TJ3=εJ3T, whe e ε∈ {−1,1}, because ±J3a e he only complex s uc u es in J
ha send e0 o Hn⊖Vφ
−. The e o e, he e exis s θ∈[0,2π) such ha
TJi=εi(cos(θ)Ji+ (−1)i+1 sin(θ)Ji+1)T, i ∈ {1,2},and TJ3=εJ3T. (4.8)
Using (4.8) and T e0=e0, we ha e
T¯
P+
1e0= cos(φ)TJ1e0+ sin(φ)TJ1e+
1
=εcos(φ)(cos(θ)J1e0+ sin(θ)J2e0)
+εsin(φ)(cos(θ)J1Te+
1+ sin(θ)J2Te+
1).
(4.9)
By P oposi ion 4.4.5, Vφ
±is p o ohomogeneous, and no e ha SO3is he only
connec ed subg oup o Sp1Spn⊂SO4n ha ac s ansi i ely and e ec i ely on he uni
sphe e o Vφ
±. Thus, we can assume ha T¯
P+
1e0=ε¯
P−
1e0, jus by composing Twi h
some elemen in he iso opy o he ac ion o SO3on Vφ
−a e0. Bu inse ing (4.9) and
¯
P−
1e0= cos(φ)J1e0+ sin(φ)J1e−
1in o he equali y T¯
P+
1e0=ε¯
P−
1e0, and analyzing
he He0and Hn⊖He0componen s (no e ha e±
1∈Hn⊖He0,Te0=e0, and T
p ese es H-o hono mali y) we ge θ= 0 and Te+
1=e−
1. Mo eo e , by (4.8) we ge
TJi=εiJiT,i∈ {1,2,3}.
Since Te0=e0,T¯
P+
1e0=ε¯
P−
1e0and TV φ
+=Vφ
−, we mus ha e T¯
P+
2e0=
±¯
P−
2e0. Then, inse ing ¯
P±
2e0= cos(φ)J2e0+ sin(φ)J2e±
2in he las equali y, and
using TJ2=J2T, we deduce ha Te+
2=e−
2. Bu his join ly wi h Te+
1=e−
1yields
a con adic ion wi h he ac ha Tis an o hogonal ans o ma ion o Hn, because
⟨e+
1, e+
2⟩ =⟨e−
1, e−
2⟩ o all φ=π/2.
4.4.2 Subspaces o dimension ou
The aim o his subsec ion is o classi y 4-dimensional eal subspaces o Hnwi h
cons an qua e nionic K¨ahle angle.
We s a by es ic ing ou a en ion o subspaces wi h φ1= 0.
P oposi ion 4.4.7. Le V⊂Hnbe a eal subspace o dimension 4wi h cons an
qua e nionic K¨ahle angle (0, φ2, φ3). Then, φ2=φ3∈[0, π/2].
P oo . Fi s o all, i φ2= 0, hen φ3= 0 by a combina ion o [13, P oposi ion 9] and
he ac ha subspaces wi h Φ(V) = (0,0, π/2) ha e dimension 3 (see Sec ion
§
4.1).
Hence, le us assume ha φ2= 0. Lemma 4.4.1 yields a basis {e0, J1e0, 2, 3}o
V, whe e i= cos(φi)Jie0+ sin(φi)Jiei, o ce ain uni ei∈Hn⊖He0,i∈ {2,3}.
The e o e, a compu a ion as in Equa ions (4.4), o each i∈ {2,3}, gi es
Ω( i)11 =cos(φ2) cos(φ3) + ⟨e2, e3⟩sin(φ2) sin(φ3)2,
Ω( i)22 = cos(φi)2+⟨J3e3, e2⟩2sin2(φ2) sin2(φ3),
Ω( i)33 = cos(φi)2+⟨J2e2, e3⟩2sin2(φ2) sin2(φ3).
78 4 Homogeneous hype su aces in HHn
Hence, by he isospec ali y o Ω,
0 = (Ω( 2)) − (Ω( 3)) = 2 cos2(φ2)−2 cos2(φ3),
om whe e we conclude φ2=φ3.
In iew o P oposi ion 4.4.7, all eal subspaces Vo Hnwi h Φ(V) = (0, φ2, φ3)
ac ually sa is y Φ(V) = (0, φ, φ). No e ha such subspaces ha e been classi ied
(see Sec ion
§
4.1).
Thus, in he ollowing esul s we will analyze he case φ1>0. We conside he
basis o Vgi en in Lemma 4.4.1.
Lemma 4.4.8. Le V⊂Hnbe a eal subspace o dimension 4such ha Φ(V) =
(φ1, φ2, φ3)wi h φ1>0. Fo each i∈ {1,3}wi h φi=π/2, we ha e ⟨ei, Jjej⟩= 0
o all j∈ {1,2,3}.
P oo . Acco ding o Lemma 4.4.1, e0has K¨ahle angle φiwi h espec o Vand
Ji∈J o each i∈ {1,2,3}. Le us ega d Hnas a complex Euclidean space C2n
whose complex s uc u e is Ji, o i∈ {1,2,3}. By [61, Theo em 2.7] he e is a
non-emp y ini e subse Ψi⊂[0, π/2] such ha V=Lφ∈ΨiVi
φis a C-o hono mal
decomposi ion o Vand Vi
φ⊂C2nis a eal subspace wi h cons an K¨ahle angle
φ∈Ψi. I ollows ha any non-ze o ∈Vi
φhas K¨ahle angle φwi h espec o V
and Ji, and he minimum ( esp. maximum) o Ψicoincides wi h he minimum ( esp.
maximum) K¨ahle angle o a non-ze o ec o ∈Vwi h espec o Vand Ji.
We claim ha φ1∈Ψ1. On he one hand, i he e exis ed φ∈Ψ1such ha
φ < φ1, hen he e would be ec o s in Vwhose K¨ahle angle wi h espec o Vand
J1∈Jis φ<φ1, hus con adic ing he minimali y o φ1by Lemma 4.1.1. On he
o he hand, i φ > φ1 o all φ∈Ψ1, hen we would ge a con adic ion wi h he ac
ha e0has K¨ahle angle φ1wi h espec o J1. Analogously, we ge ha φ3∈Ψ3.
Now assume φ1=π/2. By [61, p. 1190–1191] and he discussion abo e, we ha e
a decomposi ion V=Vφ1⊕Vψ1in o eal subspaces o cons an K¨ahle angle wi h
espec o he complex s uc u e J1, whe e ψ1∈Ψ1( he possibili y ψ1=φ1is
allowed). We also ha e ha ¯
P1:= πVφ1J1/cos(φ1) = πVJ1/cos(φ1)|Vφ1de ines a
complex s uc u e on Vφ1. As e0∈Vφ1, we ge Vφ1= spanR{e0,¯
P1e0}. Mo eo e ,
CVφ1⊥CVψ1, so Vψ1= spanR{¯
P2e0,¯
P3e0}, and o j∈ {2,3}, using Lemma 4.4.1,
0 = ⟨¯
P1e0, J1¯
Pje0⟩= sin(φ1) sin(φj)⟨e1, Jjej⟩.
Since φ1>0, we ge ⟨e1, Jjej⟩= 0. A simila a gumen wo ks o φ3, i φ3=π/2.
Be o e add essing he classi ica ion, we s a e a lemma ha e ines [60, Lemma 5.1].
Lemma 4.4.9. Assume 0< φ1≤φ2≤φ3≤π/2, and le ε∈ {−1,1}. Then, he e
exis s a subse {e1, e2, e3}o uni ec o s o R3wi h inne p oduc s
⟨ei, ei+1⟩=εcos(φi+2)−cos(φi) cos(φi+1)
sin(φi) sin(φi+1) o each i∈ {1,2,3}
4.4.2 Subspaces o dimension ou 79
i and only i cos(φ1) + cos(φ2)−εcos(φ3)≤1.
Fu he mo e, he subspace spanR{e1, e2, e3}has dimension 2i and only i cos(φ1)+
cos(φ2) + εcos(φ3)=1, and dimension 3o he wise.
P oo . A subse {e1, e2, e3}o he Euclidean space R3sa is ies he inne p oduc ela-
ions in he s a emen i and only i he associa ed G am ma ix G= (⟨ei, ej⟩)1≤i,j≤3
is posi i e semi-de ini e. This happens p ecisely when all p incipal mino s o Ga e
non-nega i e; in his p oo , by Gij we deno e he ma ix o o de 2 esul ing om
dele ing he i- h ow and he j- h column o G. Le xi:= cos(φi) o each i∈ {1,2,3}.
Hence, Gis posi i e semi-de ini e i and only i de Gii ≥0 o all i∈ {1,2,3}and
de G≥0. We compu e
de (G) = (ε+x1−x2−x3)(−ε+x1+x2−x3)(−ε+x1−x2+x3)(ε+x1+x2+x3)
(1 −x2
1)(1 −x2
2)(1 −x2
3).
Taking in o accoun ha 1 > x1≥x2≥x3≥0, one can check ha de G≥0 i
and only i −1 + x1+x2−εx3≤0. Simila ly,
de (Gii) = (1 −x2
1−x2
2−x2
3+ 2εx1x2x3)
Qj=i(1 −x2
j), i ∈ {1,2,3}.
Hence, de (Gii)≥0 i and only i
1−x2
1−x2
2−x2
3+ 2εx1x2x3≥0.(4.10)
Now, i 1 > x1≥x2≥x3≥0, one can show ha (4.10) holds p o ided ha
−1 + x1+x2−εx3≤0. This comple es he p oo o he i s claim o he lemma.
Assume ha we a e in he si ua ion o he i s asse ion o he s a emen . Then
{e1, e2, e3}spans a 3-dimensional subspace i and only i Gis posi i e de ini e, which in
his si ua ion amoun s o de G > 0. This happens p ecisely when x1+x2−εx3<1.
Hence, he p oo o he lemma will be comple e i we show ha spanR{e1, e2, e3}
canno ha e dimension 1. Assume his is he case. Then he ank o Gis 1. Hence
x1= 1 −x2+εx3, and he mino de (G33) anishes, i.e.
0 = de (G33) = −2(1 + ϵx3)
(1 + x2)(−2 + x2−ϵx3).
The e o e, x3=−ε, which yields a con adic ion. This inishes he p oo .
We a e now in posi ion o comple e he desc ip ion o 4-dimensional eal subspaces
o Hnwi h cons an qua e nionic K¨ahle angle.
P oposi ion 4.4.10. Le V⊂Hnbe a eal subspace o dimension 4and e0∈V
a uni ec o . Then Vhas cons an qua e nionic K¨ahle angle Φ(V) = (φ1, φ2, φ3),
wi h φ1>0, i and only i he e is a canonical basis {J1, J2, J3}o J,ε∈ {−1,1},
and uni ec o s e1, e2, e3∈Hn⊖He0wi h ei∈Hn⊖(Im H)ej,i,j∈ {1,2,3}, such
ha
(i) 0 < φ1≤φ2≤φ3≤π/2,

80 4 Homogeneous hype su aces in HHn
(ii) cos(φ1) + cos(φ2)−εcos(φ3)≤1,
(iii) o all i∈ {1,2,3}and indices modulo 3,
⟨ei, ei+1⟩=εcos(φi+2)−cos(φi) cos(φi+1)
sin(φi) sin(φi+1),
(i ) {cos(φi)Jie0+ sin(φi)Jiei:i= 0,1,2,3}is an o hono mal basis o V, whe e
o simplici y we pu φ0:= 0 and J0:= Id.
Mo eo e , i Vis as abo e, he K¨ahle angle o any non-ze o ∈Vwi h espec
o Jiand Vis φi, o each i∈ {1,2,3}.
P oo . In o de o p o e he necessi y, le us assume ha Vis spanned by he ba-
sis desc ibed in Lemma 4.4.1 wi h k= 4. No ice ha φ2=π/2 implies Φ(V) =
(φ, π/2, π/2); such subspaces a e classi ied (see Sec ion
§
4.1), and [13, p. 232], o-
ge he wi h some s aigh o wa d calcula ions, show ha hey can be spanned by a
basis as abo e. Thus, we can suppose φ1, φ2∈(0, π/2).
Le us i s assume φ3=π/2. A long bu elemen a y calcula ion, simila o he
one used o ob ain Equa ions (4.4), using he isospec ali y o Ω, Rema k 4.4.2 and
Lemma 4.4.8, yields
3
Y
j=1
cos2(φj) = de (Ω( ¯
Pie0))
= cos2(φi)
3
Y
j=1
j=i
(cos(φi) cos(φj) + ⟨ei, ej⟩sin(φi) sin(φj))2,
o each i∈ {1,2,3}. This implies o i∈ {1,2,3},
cos2(φi+2) = (cos (φi) cos(φi+1) + ⟨ei, ei+1⟩sin (φi) sin (φi+1)) 2.(4.11)
Using (4.11), we can also calcula e o i∈ {1,3}
3
X
j=1
cos2(φj) = (Ω( ¯
Pie0)) =
3
X
j=1
cos2(φj) + ⟨e2, Jiei⟩2sin2(φi) sin2(φ2),
which implies ⟨e2, Jiei⟩= 0, i∈ {1,3}. This, along wi h Rema k 4.4.2 and also wi h
Lemma 4.4.8, shows ha ei∈Hn⊖(Im H)ej,i, j ∈ {1,2,3}. Fu he mo e, (4.11) gi es
ise o he wo possible exp essions o ⟨ei, ei+1⟩in he s a emen (co esponding o
ε= 1 o ε=−1). No e ha such exp essions a e incompa ible o a ixed V, ha is,
i o some i∈ {1,2,3}we ha e
⟨ei, ei+1⟩=cos(φi+2)−cos(φi) cos(φi+1)
sin(φi) sin(φi+1),
⟨ei+1, ei+2⟩=−cos(φi) + cos(φi+1) cos(φi+2)
sin(φi+1) sin(φi+2),
4.4.2 Subspaces o dimension ou 81
hen one can check ha de Ω((e0+¯
Pi+1e0)/√2)= 0, which gi es a con adic-
ion wi h he assump ion φ3=π/2. Finally, he inequali y in i em (4.4.10) o he
s a emen ollows om Lemma 4.4.9.
Now assume ha φ3=π/2. Le {e0,¯
P1e0,¯
P2e0, J3e3}be he o hono mal basis
p o ided by Lemma 4.4.1. A simila compu a ion as in (4.4), using he isospec ali y
o Ω and Lemma 4.4.8, yields
2
X
i=1
cos2(φi) = (Ω( ¯
P1e0)) + (Ω( ¯
P2e0)) − (Ω(J3e3))
= cos2(φ1) + cos2(φ2) + 2 sin2(φ1) sin2(φ2)⟨J1e1, e2⟩2
+ 2 (cos(φ1) cos(φ2) + sin(φ1) sin(φ2)⟨e1, e2⟩)2.
Then,
⟨e2, J1e1⟩= 0 and ⟨e1, e2⟩=−co (φ1) co (φ2).(4.12)
Also, using (4.12), i i∈ {1,2}we ge
0 = de Ω1
√2e0+1
√2
¯
Pie0=1
4⟨e3, Jiei⟩2cos2(φ1) cos2(φ2) sin2(φi).
Thus,
⟨e3, Jiei⟩= 0, i ∈ {1,2}.(4.13)
Taking in o accoun (4.12) and (4.13), we can calcula e
2
X
i=1
cos2(φi) = (Ω( ¯
P1e0)) = cos2(φ1) + ⟨e1, e3⟩2sin2(φ1),
2
X
i=1
cos2(φi) = (Ω( ¯
P2e0)) = cos2(φ2) + sin2(φ2)(⟨e2, e3⟩2+⟨e2, J3e3⟩2),
whence ⟨e1, e3⟩=εcos(φ2)/sin(φ1),
cos2(φ1) = sin2(φ2)(⟨e2, e3⟩2+⟨e2, J3e3⟩2),(4.14)
o some ε∈ {−1,1}. Using hese ela ions we compu e
2
X
i=1
cos2(φi) = Ω1
√2
¯
P1e0+1
√2
¯
P2e0=
2
X
i=1
cos2(φi) + ε1
2⟨e2, J3e3⟩sin(2φ2),
om whe e (no e ha we a e assuming φ2=π/2)
⟨e2, J3e3⟩= 0 and ⟨e2, e3⟩=ε′cos(φ1)/sin(φ2),(4.15)
o some ε′∈ {−1,1}. Rema k 4.4.2, Lemma 4.4.8 and Equa ions (4.12), (4.13), (4.14)
and (4.15) imply ei∈Hn⊖(Im H)ej,i, j ∈ {1,2,3}. Fu he mo e, i ε′=−ε, we
82 4 Homogeneous hype su aces in HHn
ha e ha 0 is an eigen alue o Ω (e0+J3e3)/√2wi h double mul iplici y, yielding a
con adic ion wi h he ac φ2=π/2. Hence, ε′=εwhich, along wi h Lemma 4.4.9,
concludes he p oo o he necessi y in he s a emen .
The con e se implica ion ollows om e i ying by di ec calcula ion ha he
ma ix Ω( ) is diagonal wi h diagonal en ies cos2(φ1), cos2(φ2), cos2(φ3), o any
uni spanned by he basis o Vgi en in he s a emen . This also p o es he inal
claim o he p oposi ion.
Rema k 4.4.11.In iew o P oposi ion 4.4.10, he e can be ze o, one o wo ypes o
4-dimensional eal subspaces Vo Hnwi h Φ(V) = (φ1, φ2, φ3), φ1>0, depending
on whe he he iple (φ1, φ2, φ3) sa is ies cos(φ1) + cos(φ2)−cos(φ3)>1, cos(φ1) +
cos(φ2)−cos(φ3)≤1<cos(φ1)+cos(φ2)+cos(φ3), o cos(φ1)+cos(φ2)+cos(φ3)≤1,
espec i ely. Thus, i will be con enien o deno e by V+and V− he subspaces
desc ibed in P oposi ion 4.4.10 wi h ε= 1 o ε=−1, espec i ely. No e ha such
subspaces depend on he iple (φ1, φ2, φ3), bu we do no speci y his in he no a ion
o he sake o simplici y.
Obse e ha , i φ3=π/2, V+and V−a e ac ually equi alen , i.e. he e exis s
T∈Sp1Spnsuch ha TV+=V−. Indeed, one can ake a T∈Sp1Spn ha commu es
wi h Ji o all i∈ {1,2,3}, ixes each eiwi h i∈ {0,1,2}, and sends e3 o −e3. Fo
con enience, om now on we will say ha any 4-dimensional eal subspace o Hnwi h
cons an qua e nionic K¨ahle angle (φ1, φ2, π/2) is o ype V+, and no o ype V−.
In o de o encompass all examples o 4-dimensional subspaces wi h cons an
qua e nionic K¨ahle angle in o he V±-no a ion, we ha e o conside he case φ1= 0
analyzed in P oposi ion 4.4.7. Thus, we adop he con en ion ha any 4-dimensional
eal subspace Vwi h Φ(V) = (0, φ, φ), φ∈[0, π/2], is o ype V+, and no o ype
V−.
Rema k 4.4.12.The choice o he ±-no a ion in Rema k 4.4.11 is mo i a ed by ce ain
impo an p ope y o hese subspaces ha we now explain. Assume φ3=π/2. The
las claim o P oposi ion 4.4.10 enables us o ep oduce he discussion in Sec ion
§
4.3
applied o V=V±, and hence, ¯
Pi=πV±Ji/cos(φi), i∈ {1,2,3}, de e mine a Cl3-
module s uc u e on V±, which mus be i educible since dimRV±= 4. By he
classi ica ion o Cli o d modules, ei he ¯
P1¯
P2=¯
P3(and hence ¯
Pi¯
Pi+1 =¯
Pi+2 o all
i∈ {1,2,3}) o ¯
P1¯
P2=−¯
P3(and hence ¯
Pi¯
Pi+1 =−¯
Pi+2 o all i∈ {1,2,3}). One
can easily check using he basis o V±in P oposi ion 4.4.10 ha V+sa is ies p ecisely
he o me ela ion, whe eas V−sa is ies he la e .
Rema k 4.4.13.Le Vbe a eal subspace o dimension 4 in Hn,n≥4, wi h Φ(V) =
(φ1, φ2, φ3). I φ1= 0, by P oposi ion 4.4.7 we ha e Φ(V) = (0, φ, φ) o some
φ∈[0, π/2]. In his case, when φ= 0, Vis qua e nionic, i.e. V=H , o some
non-ze o ec o ∈V, whe eas i φ > 0, Vcanno i inside a qua e nionic line H,
bu can be placed in some H2(see Sec ion
§
4.1), and hus HV=H2.
Now assume φ1>0. By P oposi ion 4.4.10 and Lemma 4.4.9, Vcan be placed in
some H3i and only i V=V+and cos(φ1)+ cos(φ2)−cos(φ3) = 1, o i V=V−and
cos(φ1)+ cos(φ2) +cos(φ3) = 1; in his case HV=H3. O he wise we ha e HV=H4.
We end his sec ion by showing ha V+and V−a e no equi alen .
4.4.2 Subspaces o dimension ou 83
P oposi ion 4.4.14. The e does no exis T∈Sp1Spnsuch ha T V+=V−.
P oo . We can assume ha Φ(V+) = Φ(V−) since he qua e nionic K¨ahle angle
is p ese ed by ans o ma ions in Sp1Spn. We also assume φ3=π/2 in iew o
Rema k 4.4.11. We conside he bases o V±gi en in P oposi ion 4.4.10, whe e
we use he no a ion e±
iacco dingly, and assume wi hou loss o gene ali y ha he
canonical basis {J1, J2, J3}o Jused is he same in bo h cases.
Le us suppose ha he e is T∈Sp1Spnsuch ha TV+=V−. Deno e by π+
and π− he o hogonal p ojec ions on o V+and V−, espec i ely. By assump ion
Tπ+=π−T. Le {J′
1, J′
2, J′
3}be he canonical basis o Jgi en by J′
i=TJiT−1,
i∈ {1,2,3}, and deno e P+
i=π+Jiand P′
i=π−J′
i,i∈ {1,2,3}. Then, o any uni
ec o w∈V−and i, j ∈ {1,2,3}, we ha e
⟨P′
iw, P′
jw⟩=⟨π−TJiT−1w, π−TJjT−1w⟩=⟨T π+JiT−1w, Tπ+JjT−1w⟩
=⟨TP+
iT−1w, TP+
jT−1w⟩=⟨P+
iT−1w, P+
jT−1w⟩= cos2(φi)δij,
whe e in he las equali y we ha e used he las claim o P oposi ion 4.4.10 applied
o V+.
Thus, he canonical basis {J′
1, J′
2, J′
3}o Jdiagonalizes he bilinea o m L−
w(gi en
in P oposi ion 4.1.1) associa ed wi h he subspace V−, o any uni ec o w∈V−.
By he las claim o P oposi ion 4.4.10 applied o V−, he basis {J1, J2, J3}also has
his p ope y. Hence, he e exis s an o hogonal ma ix A∈SO3such ha
(J′
1, J′
2, J′
3)=(J1, J2, J3)A
and Acommu es wi h he diagonal ma ix wi h diagonal en ies (φ1, φ2, φ3). Then
V−coincides wi h he span o
{e−
0,cos(φ1)J′
1e−
0+ sin(φ1)J′
1e−
1,cos(φ2)J′
2e−
0+ sin(φ2)J′
2e−
2,cos(φ3)J′
3e−
0+ sin(φ3)J′
3e−
3},
whe e in his basis we ha e jus changed Jiby J′
iin he o iginal basis o V−. Since o
V−we had ¯
P−
1¯
P−
2=−¯
P−
3by Rema k 4.4.12, whe e ¯
P−
i=π−Ji/cos(φi), i∈ {1,2,3},
we also ha e ¯
P′
1¯
P′
2=−¯
P′
3, whe e ¯
P′
i=P′
i/cos(φi) = π−J′
i/cos(φi), i∈ {1,2,3}.
Howe e , deno ing ¯
P+
i=π+Ji/cos(φi), i∈ {1,2,3}, which by Rema k 4.4.12
sa is y ¯
P+
1¯
P+
2=¯
P+
3, we ob ain:
¯
P′
1¯
P′
2=1
cos(φ1) cos(φ2)π−J′
1π−J′
2=1
cos(φ1) cos(φ2)π−TJ1T−1π−TJ2T−1
=1
cos(φ1) cos(φ2)Tπ+J1π+J2T−1=T¯
P+
1¯
P+
2T−1=T¯
P+
3T−1
=1
cos(φ3)Tπ+J3T−1=1
cos(φ3)π−TJ3T−1=1
cos(φ3)π−J′
3=¯
P′
3,
which leads o a con adic ion wi h ¯
P′
1¯
P′
2=−¯
P′
3.

Chap e 5
To ally geodesic submani olds
In ui i ely, a submani old o a Riemannian mani old is o ally geodesic i i cu es
as he ambien space. I we accep he second undamen al o m as a way o mea-
su ing how complica ed he ex insic geome y o a submani old is, i u ns ou ha
o ally geodesic submani olds a e hose wi h he simples one, i.e. anishing second
undamen al o m.
To ally geodesic submani olds play a undamen al ole in Riemannian geome y.
To begin wi h, apa om hei in insic in e es , hei use has had a g ea impac ,
no only on he geome y o submani olds, bu also in a eas o geome y close o
opology, such as he s udy o spaces wi h posi i e cu a u e, o e en in numbe
heo y ( he s udy o a i hme ic g oups), as we b ie ly discuss below.
Alexand o ’s Theo em [2] s a es ha an embedded hype su ace in he Euclidean
space has cons an mean cu a u e i and only i i is a ound sphe e. This heo em
was gene alized in se e al di ec ions, see e.g. [92, 93] o some examples in he con ex
o symme ic spaces. The me hod ha Alexand o used o p o e his heo em was
o e lec a gi en hype su ace wi h cons an mean cu a u e wi h espec o o ally
geodesic hype planes and use a maximum p inciple o ellip ic ope a o s. The use o
o ally geodesic hype su aces in his si ua ion is c ucial since i gua an ees ha he
e lec ion is an isome y.
A map :N→Mbe ween (no necessa ily connec ed) mani olds is said o be
ℓ-connec ed i πi( ): πi(N)→πi(M), he induced map be ween he i- h homo opy
g oups, is an isomo phism o e e y i<ℓand a su jec ion o i=ℓ. Le Mbe a
compac Riemannian mani old wi h posi i e cu a u e and dimension n. A classical
esul in he a ea o spaces wi h posi i e cu a u e is F ankel’s Theo em, see [80],
which s a es ha wo compac o ally geodesic submani olds o Mwi h dimensions
n1and n2mus in e sec p o ided ha n1+n2≥n. Mo eo e , F ankel p o ed ha
a smoo h compac o ally geodesic embedding :N→M, whe e 2 dim N≥n, mus
be 1-connec ed, see [81]. In 2003, Wilking [184] gene alized his ac by p o ing he
so-called connec edness p inciple o mani olds wi h posi i e cu a u e. This s a es
ha i Nis a compac embedded o ally geodesic submani old o Mo codimension k,
hen he inclusion map N ,→Mis (n−2k+ 1)-connec ed. This p inciple has been
shown o cons i u e a undamen al ool o p o e igidi y esul s in spaces wi h posi i e
cu a u e, see e.g. [104, 185].
In he con ex o hype bolic geome y, o mo e gene ally, locally symme ic spaces
o non-compac ype, o ally geodesic submani olds also appea o be highly signi i-
can . Le M=G/Kbe a symme ic space o non-compac ype. Recall ha a disc e e
subg oup Γo Gis said o be a la ice i he quo ien Γ Ghas ini e olume. This
implies ha i Γac s eely on M, he space Γ Mis a locally symme ic space o
91
92 5 To ally geodesic submani olds
non-compac ype wi h ini e olume. An impo an no ion in his con ex is ha o
a i hme ic subg oup, which leads o in e es ing examples o locally symme ic spaces.
In ui i ely, a subg oup is a i hme ic i all i s poin s ha e in ege coo dina es such as
SL2(Z) in SL2(R); see [137, Chap e 5] o a p ecise de ini ion o a i hme ic la ice. Le
Γbe a la ice o SO0
1,n,M=RHn=SO0
1,n/SOn, and le N=Γ Mbe i s associa ed
locally symme ic space, whe e n≥2. Bade , Fishe , Mille , and S o e [8] p o ed
ha i Ncon ains in ini ely many p ope ly imme sed closed maximal o ally geodesic
submani olds o dimension a leas wo, hen Γis a i hme ic. They also p o ed a
simila esul in he case ha M=CHn, see [9].
Taking he p e ious discussion in o accoun , i makes sense o ca y ou a sys-
ema ic s udy o o ally geodesic submani olds in Riemannian mani olds. As we will
see, o ally geodesic submani olds a e na u ally linked o he p esence o isome ies.
Hence, he heo y o o ally geodesic submani olds is specially ich on homogeneous
spaces, and pa icula ly on symme ic ones. The s udy and classi ica ion o o ally
geodesic submani olds in gi en Riemannian mani olds will be he main goal o he
es o his hesis.
This chap e aims o gi e an o e iew o he basic concep s ela ed o o ally
geodesic submani olds in Riemannian mani olds. I is s uc u ed as ollows. In
Sec ion
§
5.1 we ecall some well-known ac s abou o ally geodesic submani olds
in Riemannian mani olds. La e , in Sec ion
§
5.2, we discuss he exis ence and he
uniqueness o o ally geodesic submani olds, and we p o e ha unde ce ain ci cum-
s ances a o ally geodesic submani old can be ex ended o a comple e one. Finally,
in Sec ion
§
5.3 we e iew some backg ound ela ed o o ally geodesic submani olds
in symme ic spaces and we discuss he mos impo an esul s conce ning o ally
geodesic submani olds in symme ic spaces.
5.1 To ally geodesic submani olds in Riemannian
mani olds
Le ¯
Mand Mbe connec ed Riemannian mani olds and :M→¯
Man isome ic
imme sion. We deno e by ¯
∇and ∇ he Le i-Ci i a connec ions o ¯
Mand M, e-
spec i ely. Recall ha :M→¯
Mis a o ally geodesic imme sion in ¯
Mi i s second
undamen al o m II anishes iden ically. The ollowing lemma exp esses a se ies o
ele an equi alences (c . [147, P oposi ion 13, p. 104]).
Lemma 5.1.1. Le Mbe a connec ed imme sed submani old o ¯
M. Then, he ol-
lowing s a emen s a e equi alen :
i) Mis o ally geodesic.
ii) I αis a cu e in Mand ∈Tα(0)M, he pa allel anspo o along αis he
same o Mand ¯
M.
iii) E e y geodesic o Mis a geodesic o ¯
M.
5.1 To ally geodesic submani olds in Riemannian mani olds 93
i ) The geodesic γ o ¯
Mwi h ini ial condi ions ˙γ(0) = pand ˙γ (0) = ∈TpM
sa is ies ha γ ( )∈M o e e y ∈(−ε, ε) o some ε > 0.
F om now on, unless o he wise s a ed, o ally geodesic submani olds will be un-
de s ood as imme sed submani olds o he ambien space.
Le us deno e by exp he exponen ial map o ¯
M. Le us conside wo o ally
geodesic submani olds M1and M2o ¯
M. Mo eo e , assume ha TpM1=TpM2 o
some p∈M1∩M2. Then, by Lemma 5.1.1, he e exis s some open neighbo hood
Uo 0 ∈TpM1=TpM2such ha expp(U)⊂M1∩M2. Mo eo e , i Miis comple e
o each i∈ {1,2}, we ha e ha M1= exppTpM1= exppTpM2=M2(since e e y
geodesic o Miis a geodesic in ¯
M). This p o es he ollowing use ul lemma.
Lemma 5.1.2. Le Mibe a o ally geodesic submani old o ¯
M, whe e i∈ {1,2}. I
TpM1=TpM2 o some p∈M1∩M2, hen M1and M2coincide a ound a neighbo hood
o p∈¯
M. Fu he mo e, i M1and M2a e comple e, hen
M1= exppTpM1= exppTpM2=M2.
Le us conside wo o ally geodesic submani olds M1and M2o ¯
Min e sec ing
a p∈M1∩M2. By Lemma 5.1.1, we can ind a small neighbo hood Uio 0 in
TpMi⊂Tp¯
Msuch ha expp(Ui)⊂Mi, o each i∈ {1,2}. Thus,
expp(U1∩U2)⊂expp(U1)∩expp(U2)⊂M1∩M2
is a chain o inclusions o open subse s o M1∩M2. This shows he ollowing.
Lemma 5.1.3. Le Mibe a o ally geodesic submani old o ¯
M, whe e i∈ {1,2}.
Then, o any p∈M1∩M2, he e is an open neighbo hood o pin M1∩M2 ha
is an embedded o ally geodesic submani old o ¯
M. In pa icula , e e y connec ed
componen o M1∩M2is a o ally geodesic submani old o ¯
M.
The nex esul ells us ha a way o cons uc o ally geodesic submani olds is by
using he isome y g oup o he ambien space ¯
M(c . [112, Chap e II, Theo em 5.1]).
Theo em 5.1.4. Le ¯
Mbe a Riemannian mani old and le S⊂Isom( ¯
M)be a subse .
Then, e e y connec ed componen o
Fix(S) := {p∈¯
M:φ(p) = p o e e y φ∈S}
is a o ally geodesic closed submani old o ¯
M.
P oo . Le p∈Fix(S) and ake V={ ∈Tp¯
M:φ∗p = o e e y φ∈S}. Now
choose a no mal neighbo hood Uo ¯
Ma ound p. We claim ha U∩Fix(S) =
expp(exp−1
p(U)∩V). No ice ha his implies ha e e y connec ed componen o
Fix(S) is an embedded submani old o ¯
M, since Vis a linea subspace o Tp¯
M.
On he one hand, le us conside a geodesic γ s a ing a p=γ(0) wi h ˙γ(0) =
∈exp−1
p(U)∩V. Thus, since ∈V, he uniqueness o geodesics, and he ac ha
isome ies map geodesics o geodesics imply ha φ◦γ=γ o e e y φ∈S. This
p o es ha expp(exp−1
p(U)∩V)⊂U∩Fix(S), since Uis a no mal neighbo hood.
94 5 To ally geodesic submani olds
On he o he hand, assume ha q∈U∩Fix(S) and ha he e is no any geodesic
γ s a ing a pwi h ini ial eloci y in exp−1
p(U)∩V eaching q. Howe e , since Uis
a no mal neighbo hood he e exis s a unique minimizing-leng h geodesic γjoining p
and q. Since γcanno ha e ini ial eloci y in exp−1
p(U)∩V, he e is some isome y
φ∈Ssuch ha φ◦γis a geodesic di e en om γbu connec ing pand q. Then, we
ge a con adic ion wi h he uniqueness o γ, and U∩Fix(S)⊂expp(exp−1
p(U)∩V).
Thus, o e e y p∈Fix(S) he e is a neighbo hood Uo p∈¯
Msuch ha U∩Fix(S)
is a submani old o ¯
M. Hence, e e y geodesic o ¯
Mwi h in ial condi ions in Fix(S)
s ays o a while in Fix(S), p o ing ha Fix(S) consis s o an union o o ally geodesic
submani olds. By de ini ion, Fix(S) is closed. The e o e, e e y connec ed componen
o Fix(S) is a o ally geodesic closed embedded submani old o ¯
M.
Al hough he abo e esul shows ha he exis ence o o ally geodesic subman-
i olds is linked o he exis ence o isome ies, he e a e o ally geodesic submani-
olds ha a e no ixed poin s o a se o isome ies. Fu he mo e, i is known ha
“gene ic” Riemannian mani olds do no admi o ally geodesic submani olds o di-
mension g ea e han 1, see [142].
Theo em 5.1.4 sugges s ha in a Riemannian mani old wi h ew isome ies he e
a e ew o ally geodesic submani olds. This implies ha an in e es ing se ing o s udy
o ally geodesic submani olds is ha o homogeneous spaces, which ha e a la ge g oup
o isome ies. A use ul cha ac e iza ion o hese spaces in e ms o Killing ields is he
ollowing.
Lemma 5.1.5. Le Mbe a connec ed Riemannian mani old. Then, he ollowing
s a emen s a e equi alen :
i) Mis homogeneous.
ii) The e exis s some poin p∈Msuch ha TpMis spanned by Killing ec o ields
o Me alua ed a p.
iii) Fo e e y p∈M,TpMis spanned by Killing ec o ields o Me alua ed a p.
P oo . Le M=G/Kbe a homogeneous space wi h educ i e decomposi ion g=k⊕p
co esponding o some poin o∈M. No ice ha pis iden i ied wi h ToM. Thus, he
Killing ec o ields induced by elemen s o p, when e alua ed a o, span ToM.
I Mis a Riemannian mani old and he e is a poin p∈Msuch ha TpMis
spanned by he Killing ec o ields e alua ed a p, hen he e exis s some open
neighbo hood Uo psuch ha e e y poin in Ulies on an in eg al cu e o a Killing
ec o ield. This implies ha he o bi o pby he ac ion o he isome y g oup is
open. Howe e , since i is also closed, we ha e ha he isome y g oup ac s ansi i ely
on M.
The nex p oposi ion shows ha o ally geodesic submani olds o homogeneous
spaces a e again homogeneous spaces.
P oposi ion 5.1.6. Le ¯
Mbe a homogeneous Riemannian mani old and le Mbe a
comple e o ally geodesic submani old o ¯
M. Then, Mis homogeneous.
5.2 On he exis ence and uniqueness o o ally geodesic submani olds 95
P oo . Le Xbe a Killing ec o ield o ¯
M. Thus, o each p∈M, we ha e he
o hogonal decomposi ion
X(p) = X(p)TpM+X(p)νpM o e e y p∈M,
whe e X(p)TpMand X(p)νpMdeno e he o hogonal p ojec ions o X(p) o TpMand
νpM, espec i ely. Since Xis a Killing ec o ield o ¯
M, o each Y∈Γ(TM), we
ha e
0 = ⟨¯
∇YX, Y ⟩=⟨¯
∇YXT M , Y ⟩+⟨¯
∇YXνM , Y ⟩=⟨∇YXT M , Y ⟩,
since Mis o ally geodesic. Thus, he angen ial p ojec ion o a Killing ec o ield
o ¯
M o M, when es ic ed o M, is a Killing ec o ield o M, since Mis comple e.
The angen space o ¯
Ma e e y poin o ¯
Mis gene a ed by Killing ields o ¯
M,
implying ha he angen space o Ma e e y poin is gene a ed by p ojec ing hese
Killing ec o ields. Hence, by Lemma 5.1.5, Mis homogeneous.
No ice ha comple e o ally geodesic submani olds o ¯
Ma e in insically homo-
geneous, bu hey a e no necessa ily ex insically homogeneous. Howe e , he con-
nec ed componen s o he ix poin se o any collec ion o isome ies a e ex insically
homogeneous submani olds, see [14, Lemma 9.1.1]. This shows once again ha he
ixed poin s se s o isome ies p o ide he mos na u al and well-beha ed examples o
o ally geodesic submani olds.
5.2 On he exis ence and uniqueness o o ally
geodesic submani olds
In his sec ion we discuss he exis ence and he uniqueness o o ally geodesic sub-
mani olds, and we p o e ha unde ce ain hypo heses a o ally geodesic submani old
can be ex ended o a comple e one.
The nex esul was p o ed by Ca an. A p oo o i can be consul ed in [14,
p. 274]. In his sec ion Bε(0) deno es he ball o adius ε > 0 a ound he o igin in a
angen space Tp¯
Mo an ambien mani old ¯
M.
Theo em 5.2.1. Le ¯
Mbe a Riemannian mani old, p∈¯
Mand Vbe a linea subspace
o Tp¯
M. Then, he e exis s a o ally geodesic submani old Mo ¯
Mwi h p∈M
and TpM=Vi and only i he e exis s some ε > 0such ha o e e y geodesic
γ: [0,1] →¯
Mwi h γ(0) = pand ˙γ(0) ∈V∩Bε(0), he Riemannian cu a u e enso
o ¯
Ma γ(1) p ese es he pa allel anspo o Valong γ om p o γ(1).
Ou in en ion o his subsec ion is o enhance Theo em 5.2.1 by p o ing a global
e sion o i , see Lemma 5.2.5. F om now on we assume ha ¯
Mis an analy ic Rie-
mannian mani old. We deno e he G assmann bundle o k-planes o T¯
Mby Gk(T¯
M)
and he injec i i y adius o ¯
Ma pby inj(p). We mainly ollow [14,
§
10.3].
Lemma 5.2.2. Le ¯
Mbe an analy ic Riemannian mani old and le Mbe a o ally
geodesic submani old o ¯
Mpassing h ough p∈¯
M. Then, expp(Bδ(0) ∩TpM)is an
embedded o ally geodesic submani old o ¯
M o e e y δ∈(0,inj(p)).

96 5 To ally geodesic submani olds
P oo . Le V=TpM. By Theo em 5.2.1, he e exis s some ε > 0 such ha o e e y
geodesic γ : [0,1] →¯
Mwi h γ (0) = pand ˙γ (0) = ∈Bε(0)∩TpM, he Riemannian
cu a u e enso o ¯
Ma γ(1) p ese es he pa allel anspo o Valong γ om p o
γ(1).
Now conside he geodesic bγ : [0,1] →¯
Mwi h bγ (0) = pand wi h ini ial eloci y
∈(Bδ(0) Bε(0)) ∩TpM. We ex end X, Y, Z ∈Vand ξ∈V⊥:= Tp¯
M⊖V o
pa allel ec o ields along bγ . Thus, he map Φ: [0,1] →R, gi en by ∈[0,1] 7→
⟨¯
R(X( ), Y ( )), Z( ), ξ( )⟩is analy ic. Howe e , by he uniqueness o pa allel ans-
po , Φ es ic ed o [0,ε
2|| ||) is iden ically ze o, which implies ha Φ is iden ically
ze o by he anali ici y o Φ. Consequen ly, he esul ollows by Theo em 5.2.1.
Le i:Mi→¯
M, wi h i∈ {1,2}, be an isome ic imme sion. We say ha 1and
2a e equi alen i he e exis s an isome y φ:M1→M2such ha 1= 2◦φ. No ice
ha e e y isome ic imme sion :M→¯
Minduces a smoo h map e
:M→Gk(T¯
M),
gi en by e
(p) = ∗p(TpM), o e e y p∈M, whe e k= dim M. The isome ic
imme sion (M, ) is said o be compa ible i Mis connec ed and e
is injec i e. Since
o ally geodesic submani olds a e locally de e mined by i s angen space a some
poin , see Lemma 5.1.2, e e y compa ible o ally geodesic isome ic imme sion is
comple ely de e mined, up o equi alence, by he image o e
in Gk(T¯
M).
Le Tbe he collec ion o all he equi alence classes o compa ible o ally geodesic
imme sions in o ¯
M. We de ine a pa ial o de ⪯in Tin he ollowing way. We w i e
(M1, 1)⪯(M2, 2) i he e exis s an injec i e local isome y i:M1→M2such ha
1= 2◦i. I his happens, we say ha (M2, 2)ex ends (M1, 1). By Lemma 5.1.2,
we ha e (M1, 1)⪯(M2, 2) i and only i e
1(M1)⊂e
1(M2).
Fo each V∈Gk(T¯
M), we deno e by FV he se o o ally geodesic imme sions
:M→¯
M om a connec ed Riemannian mani old Min o ¯
Mwi h V∈e
(M).
Mo eo e , we de ine GV:= S ∈FVe
(M)⊂Gk(T¯
M).
The ollowing lemma is a echnical one, see [14,
§
10.3] o a p oo .
Lemma 5.2.3. Following he no a ion abo e, le us assume ha GV=∅and conside
a se o compa ible o ally geodesic isome ic imme sions { i:Mi→¯
M}i∈Isuch ha
GV=Si∈Ie
i(Mi). Fu he mo e, le Y=Fi∈IMiand conside he equi alence
ela ion ∼in Ysuch ha
pi∼pji pi∈Mi,pj∈Mjand e
i(pi) = e
j(pj), whe e i, j ∈I.
Then, he ollowing s a emen s hold:
i) c
M=Y/ ∼is a connec ed smoo h mani old.
ii) The map g:c
M→¯
M,[pi]7→ i(pi)is a compa ible o ally geodesic imme sion.
iii) (c
M, g)ex ends (Mi, i) o e e y i∈I.
i ) (c
M, g)is maximal o ⪯.
5.2 On he exis ence and uniqueness o o ally geodesic submani olds 97
The isome ic imme sion g:c
M→¯
Mcons uc ed in Lemma 5.2.3 is, up o equi -
alence, he unique maximal compa ible o ally geodesic isome ic imme sion in o ¯
M
wi h V∈eg(c
M).
Lemma 5.2.4. Le Mbe a compa ible o ally geodesic submani old o an analy ic
and comple e Riemannian mani old ¯
M. Then, c
M, he ex ension o Mgi en by
Lemma 5.2.3, is comple e.
P oo . By Lemma 5.2.3, he e exis s an ex ension o (M, ) gi en by (c
M, g). We will
p o e ha (c
M, g) is comple e.
Le us p oceed by con adic ion. Thus, we will assume ha γ: [0, b)→c
Mis a uni
speed geodesic in c
M ha canno be u he ex ended beyond b > 0. Now conside
he geodesic g◦γin ¯
M. Since ¯
Mis comple e, he e exis s q= lim →b−g(γ( )) ∈¯
M.
No ice ha q∈ g(c
M). Fu he mo e, he e exis s a uni o mly no mal neighbo hood
Ua ound qin ¯
M. Then, he e is some ε > 0 such ha he injec i i y adius sa is ies
inj(p)≥ε o e e y p∈U. Now ake 0such ha b− 0< ε and q′=g◦γ( 0)∈U.
Since q′∈U, by Lemma 5.2.2, he e is a o ally geodesic submani old No ¯
Mpassing
h ough q′which also con ains q. Howe e , by Lemma 5.2.3, Nis ex ended by c
M
con adic ing he assump ion ha q∈ g(c
M).
Le kbe a non-nega i e in ege . The k- h co a ian de i a i e o he cu a u e
enso ¯
Rdeno ed by ¯
∇k¯
Ris a (1, k + 3)- enso . A subspace V⊂Tp¯
Mis in a ian
unde ( ¯
∇k¯
R)pi
(¯
∇k¯
R)(V1, . . . , Vk, X, Y, Z)∈V
o e e y X, Y, Z, V1, . . . , Vk∈V.
Theo em 5.2.5. Le ¯
Mbe an analy ic comple e Riemannian mani old, p∈¯
Mand
Va linea subspace o Tp¯
M. The e exis s a comple e o ally geodesic submani old M
o ¯
Msuch ha p∈Mand expp(V) = Mi and only i (¯
∇k¯
R)plea es Vin a ian
o e e y k≥0.
P oo . Le us ex end X, Y, Z ∈Vand ξ∈V⊥:= Tp¯
M⊖V o pa allel ec o ields
along an a bi a y geodesic γ: [0,1] →¯
Ms a ing a p∈¯
M. Then, i Vis in a ian
unde ( ¯
∇k¯
R)p o e e y k≥0 we ha e
d
d
k
| =0
¯
R(X( ), Y ( ), Z( ), ξ( )) = 0, o e e y k≥0.
By he anali ici y o ¯
M, his shows ha ¯
R(X( ), Y ( ), Z( ), ξ( )) = 0 o e e y ∈
[0,1]. Then, by Theo em 5.2.1, he e exis s a o ally geodesic submani old No ¯
M
wi h TpN=Vde ined locally a ound p∈N. Now, his o ally geodesic submani old
Ncan be ex ended o a comple e o ally geodesic submani old Mby Lemma 5.2.3
and Lemma 5.2.4.
Con e sely, i Mis a o ally geodesic submani old o ¯
M, Gauss o mula oge he
wi h Gauss and Codazzi equa ions imply ha TpMis in a ian unde ( ¯
∇k¯
R)p o
e e y k≥0 and p∈M.
98 5 To ally geodesic submani olds
5.3 To ally geodesic submani olds in symme ic
spaces
The pu pose o his sec ion is o ecall some well-known ac s and esul s ela ed o
o ally geodesic submani olds in a e y pa icula case o ambien spaces: symme ic
spaces.
Le M=G/Kbe a connec ed Riemannian symme ic space, whe e G= Isom0(M)
is he connec ed componen o he iden i y o he isome y g oup o Mand he
Lie g oup K={g∈G:g·o=o}is he iso opy a some poin o∈M. Le
gbe he Lie algeb a o G. Le Bgbe he Killing o m o g, which is de ined as
Bg(X, Y ) = (adXadY) o X, Y ∈g, whe e ad s ands o he adjoin ep esen a ion
o g.
We conside he geodesic symme y soa he base poin o∈M; i gi es ise o
an in olu i e au omo phism σo Gde ined by σ(g) = sogsowhose di e en ial a he
iden i y σ∗eis deno ed by θ. The map θis a Lie algeb a au omo phism o g, and g
decomposes as he di ec sum o ec o spaces g=k⊕p, whe e kis he ixed poin
se o θand pis he eigenspace o θco esponding o he eigen alue −1. In case
−Bg(θX, Y ) is posi i e de ini e, his spli ing is called he Ca an decomposi ion o g
wi h espec o θ, and he in olu ion θis called a Ca an in olu ion o g.
We ecall ha o any eal semisimple Lie algeb a he e exis s a Ca an in olu ion,
and any wo Ca an in olu ions in a eal semisimple Lie algeb a di e by an inne
au omo phism.
Recall om Subsec ion
§
1.3.2 ha a symme ic space is i educible i he uni e sal
co e
Mo M, which is again a symme ic space, is no isome ic o a non- i ial
p oduc o symme ic spaces. Mo eo e , a symme ic space is said o be o compac
ype, non-compac ype o Euclidean ype i Bg|p×p, he es ic ion o he Killing
o m Bg o p, is nega i e de ini e, posi i e de ini e o iden ically ze o, espec i ely.
When Mis o compac ype, hen Mis compac wi h non-nega i e sec ional cu a u e
and Gis a compac semisimple Lie g oup. I Mis o non-compac ype, hen Mis
di eomo phic o Rn, o some n≥2, and Gis a non-compac semisimple Lie g oup.
The uni e sal co e o a symme ic space spli s as a Riemannian p oduc
M=M0×M+×M−,
whe e M0, which is called he la ac o , is isome ic o a Euclidean space, and M+
and M−a e simply connec ed symme ic spaces o compac and non-compac ype,
espec i ely. I is said ha Mis semisimple i M0is a poin .
Le Σ be a connec ed o ally geodesic submani old o a symme ic space M=G/K.
By he homogenei y o M, we can assume wi hou loss o gene ali y ha o∈Σ. By
Theo em 5.2.5 and he ac ha symme ic spaces ha e pa allel cu a u e enso , a
o ally geodesic submani old Σ o Mwi h o∈Σ and V=ToΣ⊂ToMexis s i and
only i V⊂ToMis cu a u e in a ian . This means ha Ro(V, V )V⊂V, whe e R
is he Riemannian cu a u e enso o M. Recall ha using he iden i ica ion o p
and ToM, we can w i e he cu a u e enso o Ma oas
Ro(X, Y )Z=−[[X, Y ], Z], o X, Y, Z ∈ToM.
5.3 To ally geodesic submani olds in symme ic spaces 99
Thus, a subspace V⊂pis cu a u e in a ian i and only i [[X, Y ], Z]∈V o e e y
X, Y, Z ∈V. A subspace Vo pwi h his p ope y is called a Lie iple sys em in p.
Hence, he e is a one- o-one co espondence be ween Lie iple sys ems Vin pand
comple e o ally geodesic submani olds Σ in Mcon aining o∈M. In his chap e
we conside only comple e o ally geodesic submani olds since e e y o ally geodesic
submani old o a symme ic space can be ex ended o a comple e one. Fu he mo e,
i Vis a Lie iple sys em in psuch ha i s o hogonal complemen in pis also a Lie
iple sys em, we say ha Vis a e lec i e Lie iple sys em and he co esponding
o ally geodesic submani old is called e lec i e. A submani old o Mis e lec i e i
and only i i is a connec ed componen o he ixed poin se o an in olu i e isome y,
see [125].
As we saw in P oposi ion 5.1.6, comple e o ally geodesic submani olds o homo-
geneous spaces a e (in insically) homogeneous. Howe e , in he se ing o symme ic
spaces, we ha e ha comple e o ally geodesic submani olds a e ex insically homo-
geneous. To ob ain a homogeneous p esen a ion o a o ally geodesic submani old
om a Lie iple sys em we p oceed as ollows. Le V⊂pbe a Lie iple sys em in p.
De ine g′:= [V, V ]⊕V⊂k⊕p, which is clea ly a subalgeb a o gsince Vis a Lie iple
sys em. I u ns ou ha i we conside G′, he connec ed Lie subg oup o Gwi h Lie
algeb a g′, hen he G′-o bi h ough o∈Mis a o ally geodesic submani old Σ ⊂M
wi h ToΣ = V, see [14, P oposi ion 11.1.2]. This shows ha i Σ ⊂Mis a o ally
geodesic submani old passing h ough o, hen pΣ:= ToΣ is a Lie iple sys em in p
and we can de ine
kΣ:= [pΣ,pΣ],gΣ:= kΣ⊕pΣ.
Then kΣ⊂kand gΣ⊂ga e subalgeb as and i we conside he connec ed Lie
subg oups GΣ⊂Gwi h Lie algeb a gΣand KΣ⊂Kwi h Lie algeb a kΣ, hen Σ =
GΣ/KΣas homogeneous spaces. Mo eo e , e e y o ally geodesic submani old Σ o M
is in a ian unde sp o e e y p∈Σ. Thus e e y o ally geodesic submani old Σ ⊂
Mis a symme ic space wi h espec o i s induced Riemannian me ic. A o ally
geodesic submani old o a symme ic space is said o be semisimple i i is a semisimple
symme ic space. Finally, obse e ha i Vis a Lie iple sys em in p, hen iV is a
Lie iple sys em in ip, whe e iis he imagina y uni (see Subsec ion
§
1.3.2). This
means ha a o ally geodesic submani old Σ o G/Kcon aining oco esponds o a
o ally geodesic submani old Σ∗o he dual symme ic space G∗/K∗, and Σ∗is dual
o Σ. Hence, when s udying o ally geodesic submani olds i will no be es ic i e o
assume ha ou ambien symme ic space is ei he o compac ype o o non-compac
ype.
In he compac se ing, he p oblem o de e mining he opology o he sym-
me ic space ha co esponds o a pa icula Lie iple sys em is, in gene al, no
s aigh o wa d: he Lie iple sys em de e mines he symme ic space only up o
local isome y. In he non-compac se ing, howe e , we do no ha e his di icul y:
any comple e o ally geodesic submani old o a symme ic space o non-compac ype
is simply connec ed.
The p oblem o classi ying o ally geodesic submani olds in Riemannian symme ic
spaces has been a ele an and ou s anding opic o esea ch in submani old geome y
106 6 To ally geodesic submani olds in p oduc s o ank one symme ic spaces
espec i ely). Also, in HH3 he e a e wo non-cong uen o ally geodesic submani olds
homo he ic o RH3:RH3and RH3(4) ⊂RH4(4). Finally, in HH4 he e a e wo non-
cong uen o ally geodesic submani olds homo he ic o RH4:RH4and RH4(4).
Now, we will se he ollowing no a ion o he es o his sec ion. Le M=
M1×···×M , whe e Mi=Gi/Ki=FiHni(ci) is a symme ic space o non-compac
ype and ank one o each i∈ {1, . . . , }. Le o= (o1, . . . , o )∈M. Hence, we can
iden i y ToMwi h a Lie iple sys em psuch ha p=L
i=1 pi, whe e piis iden i ied
wi h ToiFiHni(ci). This implies ha gi=pi⊕[pi,pi] is he Lie algeb a o Gi.
Lemma 6.2.1. Le M=M1× ··· × M , whe e each Miis a simply connec ed,
i educible symme ic space. Le Σbe an i educible, non- la , o ally geodesic sub-
mani old o M. Then:
i) Σis k-diagonal o some k∈ {1, . . . , }in M.
Mo eo e , i Mis o non-compac ype, he ollowing s a emen s hold:
ii) The e is some pe mu a ion σo {1, . . . , }such ha Σis a o ally geodesic
submani old o Nσ(1) ×···×Nσ(k), whe e Nσ(j):= p ojσ(j)Σ,j∈ {1, . . . , }, is
a o ally geodesic submani old o Mσ(j) o e e y j∈ {1, . . . , k}.
iii) The embedding o Σis gi en by Ψ: Σ →M,Ψ(p) = (Ψ1(p),...,Ψ (p)), whe e
each map Ψj:= p ojj: Σ →Nσ(j)is a homo he y o e e y j∈ {1, . . . , k}, and
Ψl: Σ →Nσ(l)is a cons an map o e e y l∈ {k+ 1, . . . , }.
P oo . Le Σ be an i educible, non- la , o ally geodesic submani old o M. Le
p oji:gΣ→gibe he i- h o hogonal p ojec ion on o gi o i∈ {1, . . . , }. We
will p o e ha p ojiis ei he he ze o map o injec i e. Since Σ is an i educible
symme ic space, i s iso opy ep esen a ion is i educible. Fu he mo e, since Σ is
semisimple by ou assump ions, we ha e ha kΣ:= [pΣ,pΣ] is he Lie algeb a o he
iso opy g oup o Σ. Howe e , Ke p oji|pΣ⊂pΣis an in a ian subspace unde he
iso opy ep esen a ion o Σ since
p oji[Z, X] = [p ojiZ, p oji|pΣX] = 0,
whe e X∈Ke p oji|pΣand Z∈kΣ. Thus, Ke p oji= 0 o Ke p oji|pΣ=pΣ. This
implies pΣ⊂pσ(1) ⊕···⊕pσ(k) o some k∈ {1, . . . , }and some pe mu a ion σo
{1, . . . , }such ha Ke p ojσ(j)|pΣ= 0 o j∈ {1, . . . , k}and Ke p ojσ(l)|pΣ=pΣ
o l∈ {k+ 1, . . . , }. Fo a non-ze o X∈pΣ, we ha e p ojσ(j)X= 0 i and
only i j∈ {1, . . . , k}. Hence, e e y non-ze o elemen Xin pΣcan be w i en as
X=Pk
j=1 Xj, whe e each Xj∈pσ(j)is non-ze o. Thus, Σ is k-diagonal and we ha e
p o ed i).
Fu he mo e, pΣ⊂p ojσ(1) pΣ⊕···⊕p ojσ(k)pΣ, whe e each p ojσ(j)pΣis a Lie
iple sys em o pΣby Lemma 6.1.1. Mo eo e , Ke p ojσ(j)is an ideal o gΣ. Fo
e e y j∈ {1, . . . , k}we ha e Ke p ojσ(j)|pΣ= 0, which implies ha Ke p ojσ(j)= 0,
since gΣis simple as Σ is an i educible symme ic space o non-compac ype. Thus,
p ojσ(j):gΣ→p ojσ(j)gΣis a Lie algeb a isomo phism o each j∈ {1, . . . , k}.
Hence, p ojσ(j)gΣ= p ojσ(j)pΣ⊕[p ojσ(j)pΣ,p ojσ(j)pΣ] is he Lie algeb a o he

6.2 To ally geodesic submani olds in p oduc s o symme ic spaces o ank one 107
isome y g oup o Nσ(j), he o ally geodesic submani old o Mσ(j)associa ed wi h
he Lie iple sys em p ojσ(j)pΣin pσ(j). Addi ionally, aking in o accoun ha
Ke p ojσ(j)|pΣ=pΣi and only i l∈ {k+ 1, . . . , }, we ha e ha Σ p ojec s on o
a poin in Mσ(l)i and only i l∈ {k+ 1, . . . , }. The e o e, we ha e ha Σ ⊂
Nσ(1) × ··· × Nσ(k), whe e Nσ(j)is a o ally geodesic submani old o Mσ(j), which
p o es ii).
Finally, we will p o e ha p ojσ(i)is a homo he y be ween Σ and Nσ(i). Le
us ix some i∈ {1, . . . , k}and le us conside he inne p oduc in pΣgi en by
(X, Y )i:= ⟨p ojσ(i)X, p ojσ(i)Y⟩, o X, Y ∈pΣ, whe e ⟨·,·⟩ is he inne p oduc in
pinduced by i s iden i ica ion wi h ToM. Le g∈KΣ, whe e KΣis he connec ed Lie
subg oup o GΣwi h Lie algeb a kΣ. Then, g=Qk
i=1 gσ(i) o some gσ(i)∈πiKΣ,
whe e πi:G1× ··· × G →Giis he p ojec ion on o he i- h ac o . Thus, o any
X, Y ∈pΣ,
(Ad(g)X, Ad(g)Y)i=⟨p ojσ(i)Ad(g)X, p ojσ(i)Ad(g)Y⟩
=⟨p ojσ(i)Ad(gσ(i))X, p ojσ(i)Ad(gσ(i))Y⟩
=⟨Ad(gσ(i)) p ojσ(i)X, Ad(gσ(i)) p ojσ(i)Y⟩
=⟨p ojσ(i)X, p ojσ(i)Y⟩= (X, Y )i,
whe e we ha e used ha Ad(gσ(i)) is a linea isome y o ⟨·,·⟩which lea es p ojσ(i)pΣ
in a ian , since gσ(i)belongs o he iso opy o Nσ(i) o each i∈ {1, . . . , k}. Hence,
(·,·)iis a KΣ-in a ian inne p oduc in pΣ. Mo eo e , since he iso opy ep esen-
a ion o Σ is i educible by assump ion, Schu Lemma implies ha p ojσ(i)is a ho-
mo he y be ween he Lie iple sys ems pΣand p ojσ(i)pΣ o each i∈ {1, . . . , k}.
In addi ion o ha , p ojσ(i):pΣ→p ojσ(i)pΣp ese es he sec ional cu a u e
since i p ese es he Lie b acke . Thus, by [189, Theo em 1.9.2] we ha e ha
p ojσ(i): Σ →p ojσ(i)Σ is an a ine di eomo phism since Σ and p ojσ(i)Σ a e simply
connec ed. Now le p∈Σ, γbe a pa h in Σ joining oand pand eγ:= p ojσ(i)γ. Le
Pγand Peγbe he pa allel anspo s along o γand eγ, espec i ely. Since p ojσ(i)is
a ine, we ha e p ojσ(i)∗p=Peγ◦p ojσ(i)∗o◦P−1
γ o e e y p∈M. Howe e , Pγand
Peγa e isome ies and p ojσ(i)∗ois a homo he y since p ojσ(i):pΣ→p ojσ(i)pΣis a
homo he y. Thus, p ojσ(i)∗pis also a homo he y o e e y p∈Mand i u ns ou
ha p oji: Σ →p ojσ(i)Σ is a homo he y. Consequen ly, we ha e p o ed iii).
Rema k 6.2.2.Obse e ha his lemma admi s a con e se. Le M=M1× ··· ×
M , whe e each Miis an i educible symme ic space o non-compac ype. Le
Σ be a Riemannian mani old and conside he embedding Ψ: Σ →M, Ψ(p) =
(Ψ1(p),...,Ψ (p)), whe e each map Ψj: Σ →Njis ei he a homo he y o a con-
s an map, and Njis any o ally geodesic submani old o Mj,j∈ {1, . . . , }. Then,
Ψ(Σ) is a o ally geodesic submani old o N1× ··· × N , since homo he ies ca y
geodesics in o geodesics. The e o e, Ψ(Σ) is a o ally geodesic submani old o M.
Rema k 6.2.3.One impo an consequence o he p e ious esul ha dese es o
be highligh ed is he ollowing. Le Σ be an i educible, non- la , -diagonal o ally
108 6 To ally geodesic submani olds in p oduc s o ank one symme ic spaces
geodesic submani old o M=M1× ··· × M , whe e Miis an i educible symme -
ic space o non-compac ype o each i∈ {1, . . . , }. Then, wi h he usual no-
a ion, pΣ⊂p1⊕ ··· ⊕ p is an -diagonal Lie iple sys em, and we can de ine
a Lie algeb a isomo phism Φi:= p oji:gΣ→p ojigΣ, which sends pΣon o he
Lie iple sys em p ojipΣin pi. The e o e, pΣ={P
i=1 φiX:X∈p oj1pΣ},
whe e φi:= ΦiΦ−1
1: p oj1gΣ→p ojigΣis a Lie algeb a isomo phism sending
p oj1pΣon o p ojipΣ o each i∈ {1, . . . , }. Le s∈ {1, . . . , }. No ice ha
pΣs={Ps
i=1 φiX:X∈p oj1pΣ}is an s-diagonal Lie iple sys em in psuch
ha gΣsis isomo phic o gΣand hen Σs, he o ally geodesic submani old o M
co esponding o pΣs, is homo he ic o Σ.
P oposi ion 6.2.4. Le Σ1,Σ2be -diagonal, non- la , i educible, o ally geodesic
submani olds in M=M1×···×M , whe e each Miis an i educible symme ic space
o non-compac ype homo he ic o bo h Σ1and Σ2. Then, he e is g∈Isom(M1)×
···×Isom(M )⊂Isom(M)such ha gΣ1= Σ2.
P oo . Le pΣ1={Pk
i=1 φiX:X∈p1},pΣ2={Pk
i=1 ψiX:X∈p1}and gΣj:=
pΣj⊕[pΣj,pΣj], o j∈ {1,2}, whe e φi, ψi:g1→gia e Lie algeb a isomo phisms
sending p1on o pi o i∈ {2, . . . , }, and φ1=ψ1= Idg1, whe e Idg1is he iden i y
map o g1. He e we a e using Rema k 6.2.3 along wi h he assump ion ha each Mi
is an i educible symme ic space o non-compac ype homo he ic o Σ1and Σ2.
Le σi:= ψiφ−1
i∈Au (gi). Fi s o all, obse e ha σi|piis a linea isome y
o pisince σi∈Au (gi) and he inne p oduc on piis he es ic ion o he Killing
o m o gi, up o scaling. Fu he mo e, σi|pip ese es he cu a u e enso o Mi
a oisince his is gi en by Lie b acke s. Hence, σi|piis a linea isome y o pi ha
p ese es sec ional cu a u e a oi. Thus, by [189, Co olla y 2.3.14], σi|piex ends
o an isome y gi∈Isom(Mi) ha ixes oi∈Mi, since i lea es piin a ian . Then,
σi= Ad(gi) and i g:= Q
i=1 gi, whe e g1is he iden i y elemen o Isom(M1), we
ob ain
Ad(g)−1gΣ2=
Y
i=1
Ad(gi)−1gΣ2=
X
i=1
Ad(gi)−1ψiX:X∈g1
=
X
i=1
φiX:X∈g1=gΣ1,
whe e we ha e used σi=ψiφ−1
iand Ad(gi)|gj= Idgj o i=j. The e o e, he e is a
g∈Isom(M1)×···×Isom(M )⊂Isom(M) such ha gGΣ1=GΣ2g. Consequen ly,
gΣ1=gGΣ1·o=GΣ2g·o=GΣ2·o= Σ2, since g ixes o.
Lemma 6.2.5. Le Σbe a o ally geodesic submani old o M=M1×···×M , whe e
Miis a symme ic space o non-compac ype and ank one, o each i∈ {1, . . . , }.
Mo eo e , le Σ = Σ1×Σ2, whe e Σ1is i educible and no la . Then, i p ojiΣ1
and p ojjΣ2ha e posi i e dimension, we ha e ha i=j.
P oo . Le Σ = Σ1×Σ2be a o ally geodesic submani old o M, whe e Σ1is i educible
and no la . Le pΣ=pΣ1⊕pΣ2⊂p=L
i=1 pibe he co esponding Lie iple
6.2 To ally geodesic submani olds in p oduc s o symme ic spaces o ank one 109
sys em. Fix some i∈ {1, . . . , }. Le us suppose ha bpj:= p ojipΣjhas posi i e
dimension o bo h j= 1 and j= 2. By Lemma 6.2.1, we ha e ha dim bp1>1, since
Σ1is i educible and no la . Thus, we can choose X∈bp1and Y∈bp2spanning a
2-plane in pi. Mo eo e , he sec ional cu a u e sec o Miis gi en by
sec(X, Y ) = −⟨[[X, Y ], Y ], X⟩
⟨X, X⟩⟨Y, Y ⟩−⟨X, Y ⟩2.
In pa icula , [bp1,bp2]= 0, since Mihas nega i e sec ional cu a u e. Howe e , we
ha e [bp1,bp2] = [p ojipΣ1,p ojipΣ2] = p oji[pΣ1,pΣ2] = 0, since Σ is a Riemannian
p oduc o he symme ic spaces Σ1and Σ2. The e o e, we ob ain a con adic ion
wi h he assump ion ha bo h bp1and bp2ha e posi i e dimension.
Rema k 6.2.6.I is impo an o no ice ha he p e ious lemma is no ue when he
ambien space is a p oduc o i educible symme ic spaces o ank g ea e han one.
Fo ins ance, one can ind a o ally geodesic submani old Σ homo he ic o RH2×RH2
in M=M1×M2, wi h Mi=SO2,4/(SO2×SO4) o each i∈ {1,2}, such ha bo h
ac o s o Σ ha e non- i ial p ojec ion on o each ac o o M. Thus, applying [107,
Theo em 4.1 and
§
5] and duali y, he e is a o ally geodesic submani old Σ1
i×Σ2
i⊂Mi,
whe e Σ1
iand Σ2
ia e mu ually isome ic eal hype bolic planes, o each i∈ {1,2}.
Now we can conside b
Σja 2-diagonal o ally geodesic submani old in Σj
1×Σj
2 o
each j∈ {1,2}. Thus, Σ := b
Σ1×b
Σ2is a o ally geodesic submani old homo he ic o
RH2×RH2in Msuch ha bo h ac o s o Σ ha e non- i ial p ojec ion on o each
ac o o M.
P oposi ion 6.2.7. Le M1and M2be i educible symme ic spaces o compac and
non-compac ype, espec i ely. I Σ⊂M1×M2is a diagonal o ally geodesic sub-
mani old, hen Σis la .
P oo . Le Σ be a o ally geodesic submani old o M1×M2. By De-Rham Theo em,
we ha e ha he uni e sal co e ing o Σ is e
Σ = Σ0×Σ1×···×Σs, whe e Σ0is la
and each Σiis an i educible semisimple symme ic space. Mo eo e , pΣ=Ls
i=0 pΣi,
whe e pΣi⊂pΣ⊂p1⊕p2is a Lie iple sys em co esponding o he i educible
symme ic space Σi, o each i∈ {0, . . . , s}, and p1,p2a e he Lie iple sys ems
co esponding o M1and M2, espec i ely.
Le us ix some i∈ {1, . . . , s}. Then Σiis semisimple, and we ha e ha kΣi:=
[pΣi,pΣi] is he Lie algeb a o he iso opy o Σi, and gΣi:= kΣi⊕pΣiis he Lie algeb a
o he isome y g oup o Σi. Now, we de ine φij :gΣi→gj, whe e φijX= p ojjX o
each i∈ {1, . . . , s}and j∈ {1,2},gjis he Lie algeb a o he isome y g oup o Mj,
and p ojj:g1⊕g2→gjis he p ojec ion map. No ice ha Ke φij|pΣi⊂pΣiis an
in a ian subspace o he iso opy ep esen a ion o Σi. Since Σiis i educible, we
ha e Ke φij|pΣi= 0 o Ke φij|pΣi=pΣi. Mo eo e , as pΣiis diagonal by assump ion,
Ke φij|pΣi= 0 o e e y j∈ {1,2}. On he one hand, i Σiis a compac simple Lie
g oup, kΣiis simple, and hence, Ke φij|kΣi= 0 o Ke φij|kΣi=kΣi, since Ke φij|kΣi
is an ideal o kΣi. In any o he case, gΣiis simple. Then, we ha e Ke φij = 0 o
Ke φij =gΣi. Howe e , Ke φij|pΣi= 0 and his implies ha Ke φij = 0.
110 6 To ally geodesic submani olds in p oduc s o ank one symme ic spaces
To sum up, o each j∈ {1,2}, we ha e Ke φij = 0 o Ke φij =kΣi. Le us
assume ha Ke φij =kΣi o some j∈ {1,2}. In his case φijgΣi=φijpΣiis
an abelian Lie algeb a. Since gΣiis semisimple and gΣi⊂φijgΣi⊕φikgΣi, whe e
k∈ {1,2} {j}, we ha e
gΣi= [gΣi,gΣi]⊂[φijgΣi⊕φikgΣi, φijgΣi⊕φikgΣi]⊂[φikgΣi, φikgΣi]⊂φikgΣi⊂gk.
The e o e, we ob ain a con adic ion wi h he assump ion ha pΣis diagonal. Now
le us assume ha Ke φij = 0 o e e y j∈ {1,2}. This implies ha gΣiand
φijgΣia e isomo phic o e e y j∈ {1,2}. In pa icula , φijgΣiis simple o e e y
i∈ {1, . . . , s}and j∈ {1,2}, and φi1gΣiis isomo phic o φi2gΣi. Now, as φi1gΣi
is a subalgeb a o g1, we ha e ha φi1gΣiis a compac Lie algeb a. Mo eo e ,
φi2gΣi=φi2pΣi⊕[φi2pΣi, φi2pΣi], whe e φi2pΣiis a Lie iple sys em, is no a
compac Lie algeb a since i is simple and hen i is he Lie algeb a o he isome y
g oup o an i educible symme ic space o non-compac ype (see discussion abo e
Lemma 6.2.1).
Consequen ly, we ob ain a con adic ion wi h he exis ence o (non- i ial) i e-
ducible semisimple ac o s o Σ, which yields ou esul .
We now in oduce a no a ion ha will be use ul in wha ollows. Le Mand Σ
be wo symme ic spaces. We will w i e (Σ) ≤Mi Mcon ains a o ally geodesic
submani old isome ic o Σ.
A simply connec ed, educible symme ic space Mo ank 2 is a p oduc o wo
simply connec ed i educible symme ic spaces o ank one, M1and M2. Le Σ be a
o ally geodesic submani old in M=M1×M2. No ice ha i Σ is educible, hen
i has maximal ank and P oposi ion 6.1.2 implies ha Σ = Σ1×Σ2, whe e Σiis
a o ally geodesic submani old o Mi o each i∈ {1,2}. Now, i Σ is i educible,
i mus be ei he a geodesic o a semisimple o ally geodesic submani old. Le us
assume ha Σ is an i educible semisimple o ally geodesic submani old o M. Then,
by Lemma 6.2.1 i), Σ is ei he 1-diagonal o 2-diagonal. I Σ is 1-diagonal, clea ly
Σ = Σi× {pj}, whe e Σiis a o ally geodesic submani old o Miand pj∈Mj o
dis inc i, j ∈ {1,2}.
Le us assume ha M1is o compac ype and M2is o non-compac ype. By
P oposi ion 6.2.7, M=M1×M2has no diagonal o ally geodesic submani olds o
dimension g ea e han one. Hence, a 2-diagonal o ally geodesic submani old is a
geodesic.
Le us assume ha M1is la and M2is o non-compac ype. We can suppose
ha Σ is an i educible semisimple 2-diagonal o ally geodesic submani old. Thus,
e e y non-ze o ec o in pΣis o he o m X=X1+X2, whe e Xi∈piis a non-ze o
ec o in pi o each i∈ {1,2}. Howe e , since Σ is semisimple, hen pΣhas dimension
g ea e han one. Mo eo e , dim p oj1pΣ= 1. Hence, he e exis s a non-ze o ec o
X′in pΣ∩p oj2pΣ, con adic ing he assump ion ha Σ is 2-diagonal.
To sum up he abo e discussion: i M1and M2ha e opposi e ypes o one o
hem is la , hen e e y o ally geodesic submani old Σ in M=M1×M2is ei he a
geodesic o equal o Σ1×Σ2, whe e Σi⊂Miis a o ally geodesic submani old o
6.2 To ally geodesic submani olds in p oduc s o symme ic spaces o ank one 111
each i∈ {1,2}. In iew o he a gumen a ion abo e, by duali y we will assume ha
bo h ac o s in Ma e o non-compac ype. In his case, we ha e he ollowing esul .
Theo em 6.2.8. Le Mi:= FiHni(ci)be a symme ic space o non-compac ype and
ank one o i= 1,2. Gi en posi i e numbe s c′
1, c′
2, we de ine he quan i y c=c′
1c′
2
c′
1+c′
2.
Then, Σ⊂M1×M2is a o ally geodesic submani old i and only i i is equal o one
in he lis below:
i) A geodesic in M1×M2.
ii) A p oduc Σ1×Σ2, whe e Σi⊂Miis a o ally geodesic submani old o i∈ {1,2}.
iii) A o ally geodesic diagonal FHn(c), wi h F=Rand c′
i=ci, whene e (FHn(ci)) ≤
Mi o e e y i∈ {1,2}.
i ) A o ally geodesic diagonal RHn(c), wi h c′
i∈ {ci,ci
4}, whene e (RHn(c′
i)) ≤Mi
o e e y i∈ {1,2}.
Rema k 6.2.9.The diagonal embeddings in i ems iii) and i ) a e he ones desc ibed
in Lemma 6.2.1 iii). These a e o he o m Ψ: Σ →M1×M2,p∈Σ7→ (Ψ1(p),Ψ2(p)),
whe e Ψiis a homo he y be ween Σ and some o ally geodesic submani old Nio Mi
o each i∈ {1,2}(see Rema k 6.2.2).
P oo . Le us assume ha Σ has ank wo. Then, by P oposi ion 6.1.2, Σ = Σ1×Σ2,
whe e Σi⊂Miis o ally geodesic o each i= 1,2, which co esponds o i em ii) in
he s a emen .
Now assume ha Σ has ank one. Then i is ei he a geodesic, which co esponds
o i em i), o i is semisimple. In his la e case, Σ mus be isome ic o FHn(c) o
some F∈ {R,C,H,O}and c > 0. I Σ is no diagonal, i is 1-diagonal by Lemma
6.2.1 and i mus be cong uen o Σ1×{p2}o o {p1}×Σ2, whe e Σi⊂FiHni(ci) is a
o ally geodesic submani old and pi∈Mi o i= 1,2. This co esponds o i em ii) in
he s a emen . Mo eo e , i i is diagonal, i is 2-diagonal. Then, by Lemma 6.2.1 and
he classi ica ion o o ally geodesic submani olds in he ank one symme ic spaces
(see Table 6.1), we ha e ha Σ ⊂FHn(c′
1)×FHn(c′
2), o some posi i e numbe s c′
1
and c′
2. Le us u he assume ha F=R. Hence, by he classi ica ion in ank one,
c′
i=ci o i= 1,2. We will p o e ha he sec ional cu a u e o Σ sa is ies
sec(X, Y )∈−c1c2
c1+c2
,−c1c2
4(c1+c2),
o any X, Y ∈pΣspanning a 2-plane. Le p′
i:= p ojipΣbe a Lie iple sys em
associa ed wi h ToiFHn(ci), whe e oi∈FHn(ci). Mo eo e , conside he Lie algeb a
o he isome y g oup o FHn(ci), which is g′
i:= p′
i⊕[p′
i,p′
i]. Then, he Lie iple sys em
co esponding o Σ is pΣ={X1+φX1:X1∈p′
1} o some Lie algeb a isomo phism
φ:g′
1→g′
2 ha sends p′
1on o p′
2(see Rema k 6.2.3). Now le ⟨·,·⟩1be he me ic
o FHn(1), and sec1(·,·) i s sec ional cu a u e. We can ega d he induced me ic o

112 6 To ally geodesic submani olds in p oduc s o ank one symme ic spaces
FHn(ci) on p′
ias a posi i e mul iple o ⟨·,·⟩1. Hence, we can w i e he me ic o Σ a
(o1, o2) as
⟨·,·⟩ := λ1⟨p oj1·,p oj1·⟩1+λ2⟨p oj2·,p oj2·⟩1,
o some λ1, λ2>0. Le X=X1+φX1, Y =Y1+φY1∈pΣ, whe e X1, Y1∈p′
1
sa is y ⟨X1, X1⟩1=⟨Y1, Y1⟩1= 1 and ⟨X1, Y1⟩1= 0. Mo eo e , we ha e
⟨[[X, Y ], Y ], X⟩=λ1⟨[[X1, Y1], Y1], X1⟩1+λ2⟨[[φX1, φY1], φY1], φX1⟩1
=λ1⟨[[X1, Y1], Y1], X1⟩1+λ2⟨φ[[X1, Y1], Y1], φX1⟩1
=−(λ1+λ2) sec1(X1, Y1),
⟨X, X⟩=λ1⟨X1, X1⟩1+λ2⟨φX1, φX1⟩1=λ1+λ2,
⟨Y, Y ⟩=λ1⟨Y1, Y1⟩1+λ2⟨φY1, φY1⟩1=λ1+λ2,
⟨X, Y ⟩=λ1⟨X1, Y1⟩1+λ2⟨φX1, φY1⟩1= 0,
whe e we ha e used ha φp ese es ⟨·,·⟩1since φp ese es he Killing o m. Thus,
he sec ional cu a u e a (o1, o2)∈Σ o he 2-plane spanned by {X, Y }is gi en by
sec(X, Y ) = sec1(X1, Y1)
λ1+λ2
=c1c2
c1+c2
sec1(X1, Y1)∈−c1c2
c1+c2
,−c1c2
4(c1+c2),
since λi= 1/ci o each i∈ {1,2}, because λg has sec ional cu a u e 1
λsec, when g
is a Riemannian me ic, sec i s sec ional cu a u e and λa posi i e numbe . Hence,
Σ mus be isome ic o a diagonal FHn(c1c2
c1+c2) whene e (FHn(ci)) ≤Mi o e e y
i∈ {1,2}and F=R. This co esponds o i em iii) in he s a emen .
Now le us assume ha F=R. Again, by Lemma 6.2.1 and he classi ica ion
o o ally geodesic submani olds in symme ic spaces o non-compac ype and ank
one (see Table 6.1), we ha e Σ ⊂RHn(c′
1)×RHn(c′
2), whe e c′
i∈ {ci, ci/4}is such
ha (RHn(c′
i)) ≤Mi o e e y i∈ {1,2}. A simila compu a ion as abo e yields ha
he sec ional cu a u e o Σ is equal o −c′
1c′
2
c′
1+c′
2. Then, Σ is isome ic o RHn(c′
1c′
2
c′
1+c′
2),
which co esponds o i em i ) in he s a emen .
Rema k 6.2.10.No ice ha unlike in he i educible ank one case, i Mis a educible
space o ank wo hen we can ind mu ually isome ic diagonal o ally geodesic sub-
mani olds Σ1and Σ2which a e no cong uen in M. This implies ha he hypo hesis
o Mibeing homo he ic o Σ1and Σ2 o each i∈ {1,2}in P oposi ion 6.2.4 is c ucial.
Le us conside M=M1×M2, whe e M1=CH2and M2=CH3. Le us assume ha
he e is some φ∈Isom(M) such ha φΣ1= Σ2whe e
Σ1:= RH2(4/5) ⊂L1:= RH2(4) ×RH2⊂CH2×CH2⊂M1×M2,
Σ2:= RH2(4/5) ⊂L2:= RH2×RH2(4) ⊂CH2×CH2⊂M1×M2.
Clea ly, each isome y o M1×M2mus p ese e bo h ac o s since M1and M2a e
no isome ic. Then, we ha e Σ2⊂φL1∩L2. Howe e , since RH2(4) and RH2a e
complex and o ally eal submani olds in CH2, espec i ely, we ha e φL1∩L2⊂R×R,
whe e R×Ris a maximal o ally geodesic la submani old o M1×M2. Mo eo e ,
6.2 To ally geodesic submani olds in p oduc s o symme ic spaces o ank one 113
since he in e sec ion o o ally geodesic submani olds is o ally geodesic, his implies
ha Σ2⊂φL1∩L2⊂R×R⊂M1×M2, which con adic s he ac ha Σ2is no
la and p o es ha such φcanno exis .
Le us ecall ha he elemen a y symme ic polynomial eko o de k∈ {0, . . . , n}
in n a iables is de ined as ek(X1, . . . , Xn) = P1≤i1<...<ik≤nXi1···Xik. Then, we
ha e he ollowing gene aliza ion o Theo em 6.2.8 o he case o diagonal o ally
geodesic submani olds in a bi a y p oduc s o symme ic spaces o ank one.
Co olla y 6.2.11. Le M=M1×···×M , whe e each Mi=FiHni(ci)is a symme ic
space o non-compac ype and ank one o i∈ {1, . . . , }. Gi en posi i e numbe s
{c′
i}
i=1, we de ine he quan i y
c:= Q
i=1 c′
i
e −1(c′
1, . . . , c′
),
whe e e −1is he elemen a y symme ic polynomial o deg ee −1in a iables.
I Σis a non- la , i educible, -diagonal, o ally geodesic submani old in M, hen
i is isome ic o one in he lis below:
i) RHn(c), wi h c′
i∈ {ci,ci
4}, whene e (RHn(c′
i)) ≤Mi o e e y i∈ {1, . . . , }.
ii) FHn(c), wi h F=Rand c′
i=ci, whene e (FHn(ci)) ≤Mi, o e e y i∈
{1, . . . , }.
P oo . Le Σ ⊂Mbe a non- la , -diagonal, i educible, o ally geodesic submani old.
We will p oceed by induc ion on . The s a emen is ue o = 1 by he classi ica ion
o o ally geodesic submani olds in symme ic spaces o non-compac ype and ank
one (see Table 6.1). Le us assume ha i is ue o −1 ac o s and we will p o e
i o . Obse e ha Σ is con ained in b
Σ×b
Σ′whe e b
Σ is he p ojec ion o Σ on o
M1×···×M −1and b
Σ′is he p ojec ion o Σ on o M . By Lemma 6.1.1, we ha e
ha b
Σ and b
Σ′a e o ally geodesic submani olds o M. Fu he mo e, by Lemma
6.2.1 iii) and Rema k 6.2.3, we ha e ha b
Σ and b
Σ′a e homo he ic o Σ. Mo eo e ,
b
Σ is ( −1)-diagonal in M, since Σ is -diagonal in M, and by induc ion hypo hesis,
b
Σ is isome ic o FHn(ec) o F∈ {R,C,H,O}, whe e
ec=Q −1
i=1 c′
i
e −2(c′
1, . . . , c′
−1),
wi h c′
i∈ {ci, ci/4} o each i∈ {1, . . . , −1}. Le us assume ha F=R, since
he esul ollows simila ly in he o he cases. Then, by Theo em 6.2.8, since Σ is
2-diagonal in b
Σ×b
Σ′, and b
Σ and b
Σ′a e o ank one, he sec ional cu a u e o Σ is
equal o he opposi e o
c=ec c′
ec+c′
=Q
i=1 c′
i
e −1(c′
1, . . . , c′
),
whe e we ha e used e −1(X1, . . . , X ) = e −1(X1, . . . , X −1)+e −2(X1, . . . , X −1)X ,
o a bi a y a iables X1, . . . , X .
114 6 To ally geodesic submani olds in p oduc s o ank one symme ic spaces
Now we will p o ide he classi ica ion o o ally geodesic submani olds in p oduc s
o symme ic spaces o ank one by in oducing a combina o ial objec ha we call
adap ed Young ableau.
We i s ecall he well-known no ions o pa i ion o a posi i e in ege and o
Young diag am. Le ≥1 be a posi i e in ege . Then, a pa i ion o is a ec o
λ= (λ1, . . . , λk)∈Zksuch ha =Pk
j=1 λjand λ1≥. . . ≥λk≥1. Fo each
pa i ion λwe associa e a Young diag am. This is a collec ion o boxes wi h λjboxes
in he j- h ow, o each j∈ {1, . . . , k}.
Now we will in oduce he no ion o Young ableau adap ed o a p oduc M
o symme ic spaces o non-compac ype and ank one. Le us conside M=
F1Hn1(c1)× ··· × F Hn (c ), whe e ni≥2, ci>0 and Fi∈ {R,C,H,O} o each
i∈ {1, . . . , }. Le λ= (λ1, . . . , λk) be a pa i ion o and le us conside i s Young
diag am. We will add o he m- h box in he j- h ow a o ally geodesic inclusion
F′
ij,m Hn′
ij,m (c′
ij,m )⊂Fij,m Hnij,m (cij,m ), o e e y j∈ {1, . . . , k}and m∈ {1, . . . , λj},
whe e Fk
j=1{ij,1, . . . , ij,λj}={1, . . . , }. Fu he mo e, we equi e all o ally geodesic
submani olds appea ing in he j- h ow o be mu ually homo he ic. A Young diag am
wi h his in o ma ion will be called Young ableau adap ed o M. See Figu e 6.1 o
a ious examples o his.
RH3(c1)⊂RH3(c1)RH3(c2/4) ⊂CH3(c2)RH3(c3)⊂HH3(c3)RH3c1c2c3
c1c2+4c1c3+c2c3
CH2(c2)⊂CH3(c2)CH2(c3)⊂HH3(c3)CH2c2c3
c2+c3
RH2(c1)⊂RH3(c1)RH2(c1)
RH3(c1)⊂RH3(c1)RH3(c1)
RH3(c2/4) ⊂CH3(c2)RH3(c2/4)
RH4(c3)⊂HH3(c3)RH4(c3)
Figu e 6.1: Th ee examples o Young ableaux adap ed o he p oduc M=RH3(c1)×
CH3(c2)×HH3(c3), along wi h he i educible ac o s o he co esponding o ally
geodesic submani olds p esen ed a he end o each ow (see P oposi ion 6.2.12).
No ice ha he isome y ype o hese o ally geodesic submani olds can be compu ed
using Co olla y 6.2.11.
P oposi ion 6.2.12. Le M=M1×···×M , whe e Miis a symme ic space o non-
compac ype and ank one o each i∈ {1, . . . , }. Then, he ollowing s a emen s
hold:
i) Fo each Young ableau Tadap ed o Mwe can a ach a se S(T)o semisimple
o ally geodesic submani olds ΣTo M ha ha e non- i ial p ojec ion on o each
ac o o M.
ii) I ΣTand e
ΣTbelong o S(T), hen ΣTis isome ic o e
ΣT.
iii) I Σis a semisimple o ally geodesic submani old o M ha has non- i ial
6.2 To ally geodesic submani olds in p oduc s o symme ic spaces o ank one 115
p ojec ion on o each ac o o M, hen i is equal o some ΣT∈ S(T) o some
Young ableau Tadap ed o M.
P oo . Fi s o all, we will see how o cons uc a o ally geodesic submani old o M
om a Young ableau adap ed o M. Le Tbe a Young ableau adap ed o Mand
le us assume ha i has k ows. Le us u he assume ha i has λjboxes in he
j- h ow. Namely,
F′ij,1Hn′
ij,1(c′
ij,1)⊂Fij,1Hnij,1(cij,1), . . . , F′ij,λjHn′
ij,λj(c′
ij,λj)⊂Fij,λjHnij,λj(cij,λj)
a e he labels in he boxes in he j- h ow, whe e we ha e Fk
j=1{ij,1, . . . , ij,λj}=
{1, . . . , }. Le pij,m be a Lie iple sys em co esponding o Mij,m =Fij,m Hnij,m (cij,m ),
o each m∈ {1, . . . , λj}. Then, o each m∈ {1, . . . , λj}, he e is some Lie iple
sys em p′
ij,m ⊂pij,m ha co esponds o he o ally geodesic embedding in he m- h
box o he j- h ow o T. No ice ha , by cons uc ion o T, hese o ally geodesic
submani olds a e mu ually homo he ic. Le us de ine g′
ij,m := p′
ij,m ⊕[p′
ij,m ,p′
ij,m ].
Clea ly, o any ixed j∈ {1, . . . , k}, all hese Lie algeb as g′
ij,m ,m∈ {1, . . . , λj},
a e mu ually isomo phic because hey a e he Lie algeb as o he isome y g oups
o mu ually homo he ic semisimple symme ic spaces. Le φj
1,m :g′
ij,1→g′
ij,m be
a Lie algeb a isomo phism sending p′
ij,1in o p′
ij,m o m∈ {2, . . . , λj}, and φj
1,1be
he iden i y map o g′
ij,1. Now, we de ine bpj:= nPλj
m=1 φj
1,mX:X∈p′
ij,1o o each
j∈ {1, . . . , k}. Then, bpjis a Lie iple sys em in p ha is λj-diagonal.
We pe o m his p ocess o each ow j∈ {1, . . . , k} o de ine pT:= Lk
j=1 bpj. By
cons uc ion, we ha e ha [bpj,bpj′] = 0 o j=j′in {1, . . . , k}. Hence, pTis a Lie
iple sys em in pand we will deno e by ΣT= Σ1×···×Σki s co esponding o ally
geodesic submani old, whe e Σjis he o ally geodesic submani old co esponding
o bpj. Consequen ly, o a Young ableau Tadap ed o M, we ha e cons uc ed a
semisimple o ally geodesic submani old ΣTo M, ha has non- i ial p ojec ion on o
each ac o o M. Howe e , no ice ha his cons uc ion depends on he Lie iple
sys em p′
ij,m in he subspace pij,m and on he Lie algeb a isomo phism φj
1,m ha we
chose, and i we choose di e en Lie iple sys ems and isomo phisms, we ge di e en
semisimple o ally geodesic submani olds wi h non- i ial p ojec ion on o each ac o
o M. Hence, o each Young ableau Tadap ed o Mwe can a ach a se S(T) which
is equal o he se o all he o ally geodesic submani olds ha can be cons uc ed
om T h ough he p ocess desc ibed abo e. This p o es i).
Now we will check ha all o ally geodesic submani olds in S(T) a e mu ually
isome ic. Fi s o all, obse e ha wo o ally geodesic submani olds o a symme ic
space o ank one a e cong uen i and only i hey a e isome ic. Hence, he o ally
geodesic embedding F′
ij,m Hn′
ij,m (c′
ij,m )⊂Fij,m Hnij,m (cij,m ) ep esen s a cong uence
class o o ally geodesic embeddings in Mij,m =Fij,m Hnij,m (cij,m ). Le Nij,m and
e
Nij,m be cong uen o ally geodesic submani olds in Mij,m co esponding o he in-
clusion in he m- h box o he j- h ow o each m∈ {1, . . . , λj}and j∈ {1, . . . , }.
Then, he e is a isome y φij,m o Mij,m such ha φij,m Nij,m =e
Nij,m . Following he
122 7 To ally geodesic submani olds in excep ional symme ic spaces
o he ele an subalgeb as, we cons uc a se which con ains all o hem. Then i
emains o decide which o hese subalgeb as gi e ise o maximal o ally geodesic
submani olds. In o de o do so, we de elop some c i e ia o maximali y.
We would like o ema k ha ou me hods can be used o classi y maximal o ally
geodesic submani olds in symme ic spaces whose isome y g oup has ank less o
equal han eigh . The e o e, one could lis all maximal o ally geodesic submani olds
up o isome y in symme ic spaces wi h isome y g oup o ank less o equal han
eigh . Howe e , in his hesis, we con en ou sel es wi h lis ing he classi ica ion in
he excep ional symme ic spaces.
Onishchik [148] in oduced an in a ian o symme ic spaces conce ning o ally
geodesic submani olds called index, which is de ined as he minimal codimension o a
o ally geodesic p ope submani old. Be nd , Olmos and Rod ´ıguez [20, 21, 22, 23, 24]
ha e compu ed he index i(M) o e e y i educible symme ic space M. In pa icula ,
hey p o ed wha hey called he Index Conjec u e [20]. This conjec u e s a es ha
in an i educible symme ic space o non-compac ype M=G2
2/SO4, he e is some
e lec i e submani old Σ o Mwhose codimension equals he index o M.
Gene alizing a no ion in oduced by Dynkin [74], in his chap e we de ine he
Dynkin index o ce ain semisimple subalgeb as o simple eal Lie algeb as, see De -
ini ion 7.4.2. We use his o cha ac e ize he isome y ypes o o ally geodesic em-
beddings o semisimple symme ic spaces in o i educible symme ic spaces. This
cha ac e iza ion allows us o de i e a esul analogous o he Index Conjec u e:
Theo em B. Le Mbe an i educible symme ic space o non-compac ype. Then,
he e is some o ally geodesic submani old Σin Mwi h i(M) = codim(Σ) such ha
he Dynkin index o he semisimple pa o he Lie algeb a o he isome y g oup o Σ
equals (1,1, . . . , 1).
This chap e is o ganized as ollows. We e isi di e en o mula ions o he
Ka pele ich-Mos ow heo em in Sec ion
§
7.1. Some use ul ac s abou he complex-
i ica ion o Lie subalgeb as o a eal Lie algeb a a e ecalled in Sec ion
§
7.2. In Sec-
ion
§
7.3, we p o e a co espondence be ween maximal semisimple o ally geodesic
submani olds in symme ic spaces o non-compac ype and ce ain subalgeb as o
he isome y algeb a o he ambien space. Then we specialize ou s udy o o ally
geodesic submani olds o symme ic spaces o he se ing o excep ional symme ic
spaces. In Sec ion
§
7.4, we in oduce an in a ian o ce ain semisimple o ally
geodesic submani olds in symme ic spaces o non-compac ype ha we call Dynkin
index, which cha ac e izes hese submani olds up o isome ies. In Sec ion
§
7.5, we
classi y maximal o ally geodesic submani olds in excep ional symme ic spaces whose
isome y g oup is absolu ely simple, and in Sec ion
§
7.6 we deal wi h he case when
he isome y g oup is no absolu ely simple. Sec ion
§
7.7 con ains he p oo s o he
wo main esul s (Theo em A and B abo e).

7.1 Ka pele ich-Mos ow Theo em 123
7.1 Ka pele ich-Mos ow Theo em
A undamen al esul in he s udy o o ally geodesic submani olds in symme ic
spaces o non-compac ype is known as he Ka pele ich Theo em [103], see also [138]
and [66].
Theo em 7.1.1. Le M=G/Kbe a symme ic space o non-compac ype. Then
any connec ed semisimple subg oup H⊂Gac s on Mwi h a o ally geodesic o bi .
An equi alen , mo e algeb aic o mula ion, see [150, Co olla y 1, p. 46], is he
ollowing.
Theo em 7.1.2. Le :h→gbe a homomo phism o eal semisimple Lie algeb as
and le a Ca an decomposi ion h=k′⊕p′be gi en. Then he e exis s a Ca an
decomposi ion g=k⊕psuch ha (k′)⊂kand (p′)⊂p.
A subalgeb a ho a eal semisimple Lie algeb a gis called canonically embedded
in gwi h espec o some Ca an decomposi ion g=k⊕pi h= (h∩k)⊕(h∩p). This
is equi alen o hbeing θ-in a ian , whe e θis he Ca an in olu ion associa ed wi h
he decomposi ion g=k⊕p.
An algeb aic g oup o e K∈ {R,C}is an a ine algeb aic a ie y Go e Kendowed
wi h a g oup s uc u e o which he map G×G→G, (x, y)7→ xy−1is polynomial. I
u ns ou ha an algeb aic g oup o e K∈ {R,C}is a Lie g oup o e K∈ {R,C}. An
algeb aic subg oup o an algeb aic g oup Gis a closed subg oup o G(in he Za iski
opology). An algeb aic subg oup is i sel an algeb aic g oup. A Lie algeb a gis
algeb aic i i is he Lie algeb a o some i educible algeb aic subg oup Go GLn(K),
wi h K∈ {R,C}. In pa icula , semisimple Lie algeb as a e algeb aic, see [151,
p. 138]. Le Gbe an algeb aic g oup o e K∈ {R,C}wi h Lie algeb a g. A Lie
subalgeb a ho gis called algeb aic i he e exis s an algeb aic subg oup Ho Gwi h
Lie algeb a h. Fo any Lie subalgeb a ho g he e is a smalles algeb aic subalgeb a ha
o gcon aining h, called he algeb aic closu e o hin g. An algeb aic subalgeb a ho a
complex semisimple Lie algeb a gis said o be educ i e i i is a educ i e Lie algeb a
and i s cen e consis s o semisimple elemen s, i.e. o elemen s X∈h o which he
linea map adXis a diagonalizable endomo phism o he ec o space h.
Rema k 7.1.3.Le gbe a semisimple Lie algeb a. By [152, Chap e 1,
§
6.2, Theo-
em 6.2], i his a subalgeb a o g, hen [h,h]=[ha,ha]. Le lbe a maximal p ope Lie
subalgeb a o g. Then [la,la]=[l,l]=g. Hence la=g, and hus by maximali y la=l.
Hence a maximal p ope Lie subalgeb a o a semisimple Lie algeb a is algeb aic.
A mo e gene al e sion o Theo em 7.1.2 can be o mula ed as ollows, see [152,
Theo em 3.6, Chap e 6].
Theo em 7.1.4 (Ka pele ich-Mos ow).An algeb aic subalgeb a o a eal semisimple
Lie algeb a gis educ i e i and only i i is canonically embedded in gwi h espec o
some Ca an decomposi ion o g.
124 7 To ally geodesic submani olds in excep ional symme ic spaces
7.2 Complexi ica ion o subalgeb as
Le gbe a eal Lie algeb a. Recall ha i s complexi ica ion is de ined by gC:= g⊗RC.
Con e sely, he eali ica ion hRo a complex Lie algeb a his de ined as he eal Lie
algeb a ob ained om hby es ic ing he scala s o he eals. Fu he mo e, i g
is a complex Lie algeb a, hen a subalgeb a g0o gRis called a eal o m o gi
g=g0+ig0and g0∩ig0= 0, whe e iis he imagina y uni . We say ha a Lie
algeb a is o non-compac ype i i is semisimple and all o i s simple ideals a e non-
compac Lie algeb as. We say ha a subalgeb a o a Lie algeb a gis maximal o
non-compac ype i i is maximal among all p ope subalgeb as o non-compac ype
o g.
A eal Lie algeb a gis called absolu ely simple i i is simple and i s complexi ica ion
gCis a simple complex Lie algeb a. In case gis an absolu ely simple eal Lie algeb a,
gis a eal o m o he simple complex Lie algeb a gC. In case gis a simple, bu
no absolu ely simple, eal Lie algeb a, gis he eali ica ion o a simple complex Lie
algeb a.
Rema k 7.2.1.In la e sec ions o his chap e , we do no some imes dis inguish in
ou no a ion be ween a complex Lie algeb a and i s eali ica ion when i is clea om
he con ex which Lie algeb a s uc u e we conside .
Rema k 7.2.2.Recall om he classi ica ion o Riemannian symme ic spaces ha
he e a e wo classes o i educible Riemannian symme ic spaces o non-compac
ype. I Gis he connec ed componen o he isome y g oup o such a M=G/K, we
dis inguish be ween he ollowing wo cases:

Symme ic spaces o ype III :gis absolu ely simple.

Symme ic spaces o ype IV :gis simple, bu no absolu ely simple.
Lemma 7.2.3. Le gbe a simple eal Lie algeb a and le h⊂gbe a subalgeb a. Then
he ollowing s a emen s hold:
i) I hC⊂gCis a maximal educ i e algeb aic subalgeb a, hen h⊂gis a maximal
educ i e algeb aic subalgeb a.
ii) I hC⊂gCis a maximal semisimple subalgeb a, hen h⊂gis a maximal semisim-
ple subalgeb a.
P oo . By [150,
§
2, P oposi ion 2(i)], hCis a semisimple Lie algeb a i and only i his
a semisimple Lie algeb a. Thus, [h,h] is semisimple i and only i [h,h]Cis semisimple.
Since [h,h]C⊕Z(h)C= [hC,hC]⊕Z(hC), his a educ i e Lie algeb a i and only i hC
is a educ i e Lie algeb a. Mo eo e , adZ(h)consis s o semisimple elemen s i and
only i adZ(hC)does. Consequen ly, we ha e p o ed ha h⊂gis a educ i e algeb aic
subalgeb a i and only i hC⊂gCis a educ i e algeb aic subalgeb a.
Le us assume ha hCis a maximal educ i e algeb aic ( esp. semisimple) subal-
geb a o gC, and ha he e is some educ i e algeb aic ( esp. semisimple) subalgeb a
l⊊gsuch ha h⊂l⊂g. Then hC⊂lC⊂gCand lCis educ i e ( esp. semisimple).
7.2 Complexi ica ion o subalgeb as 125
Since hCis a maximal educ i e algeb aic ( esp. maximal semisimple) subalgeb a, we
ha e ha hC=lC. The e o e, hand lha e he same dimension and we conclude ha
h=l.
We say ha a subalgeb a h⊂go a complex semisimple Lie algeb a gis a
egula subalgeb a i his no malized by some Ca an subalgeb a a⊂g. Following
Dynkin [74], we say ha a subalgeb a h⊂gis an R-subalgeb a i i is con ained in a
egula p ope subalgeb a, and we say ha i is an S-subalgeb a i i is no con ained
in a egula p ope subalgeb a. I gis no absolu ely simple, i s maximal subalgeb as
o non-compac ype a e desc ibed by Lemma 7.2.4, which is simila o [73, Appendix
o Chap e 1, Theo em 1.6].
Lemma 7.2.4. Le gbe a simple complex Lie algeb a and h⊂gRbe a subalgeb a ha
is maximal among he subalgeb as o non-compac ype. Then hcoincides wi h one o
he ollowing:
i) a maximal semisimple egula subalgeb a o g,
ii) a maximal S-subalgeb a o g,
iii) a non-compac eal o m o g.
Con e sely, all o he abo e a e maximal subalgeb as o non-compac ype o gR.
P oo . Le hbe a subalgeb a ha is maximal among he subalgeb as o non-compac
ype o gR. Fo all subalgeb as o gRwe ha e ha h+ihis a complex subalgeb a
o gand h0:= {X∈h:λX ∈h o all λ∈C}=h∩ihis an ideal o h+ih, see [73,
Appendix o Chap e 1]. I h+ih=g, hen i ollows ha h0is an ideal o g. Since
gis simple, i ollows ha h0= 0 and hence ha his a eal o m o g. Ob iously, h
is no a compac eal o m.
Now we may assume ha h+ih=g. Then h+ihis con ained in some maximal
subalgeb a eho g. Conside i s he case whe e he e is a maximal egula educ i e
subalgeb a ehcon aining h+ih. I he subalgeb a eho he complex Lie algeb a gis
semisimple, hen i is au oma ically a subalgeb a o non-compac ype o gR. Hence,
in his case we ha e h=ehand his one o he subalgeb as gi en in [74, Table 12]. On
he o he hand, i he subalgeb a eho gis non-semisimple, hen h+ihis con ained
in one o he subalgeb as in [74, Table 12a] and since hese a e maximal semisimple
egula subalgeb as and also subalgeb as o non-compac ype in gR, by maximali y,
his conjuga e o one o hem.
I emains he case when h+ihis no con ained in a egula subalgeb a o g. Then
i is con ained in a maximal S-subalgeb a o g. I ollows om [74, Theo em 7.3],
ha e e y non-semisimple subalgeb a o a complex semisimple Lie algeb a is an R-
subalgeb a. The e o e we know ha S-subalgeb as o ga e semisimple and hence
hey a e also subalgeb as o non-compac ype in gR. By maximali y, i ollows now
ha his a maximal S-subalgeb a o gin his case.
Le us now show ha he con e se holds. Since he eal o ms o ga e simple,
hey a e ei he compac o o non-compac ype. The subalgeb as in i) and, by [74,
No. 24, Theo em 7.3], in ii) a e maximal semisimple subalgeb as o gand hey a e o
non-compac ype, hence maximal subalgeb as o non-compac ype.
126 7 To ally geodesic submani olds in excep ional symme ic spaces
7.3 Maximal semisimple o ally geodesic submani-
olds
We p o e below a heo em ha es ablishes a one- o-one co espondence be ween max-
imal semisimple o ally geodesic submani olds in symme ic spaces o non-compac
ype and ce ain subalgeb as o he Lie algeb a o he isome y g oup o he ambien
space.
A esul o Aleksee sky and Di Scala [4, P oposi ion 5.5] s a es ha in a symme ic
space o non-compac ype M=G/K, i he ac ion o a subg oup o Ghas wo o ally
geodesic o bi s, hey a e isomo phic as homogeneous spaces. Imp o ing on his esul ,
we p o e he ollowing s a emen which p o ides uniqueness up o cong uence o he
exis ence s a emen in Theo em 7.1.1. Howe e , we no e ha , in his esul , he g oup
ac ing upon is no assumed o be semisimple.
P oposi ion 7.3.1. Le M=G/Kbe a symme ic space o non-compac ype and
H⊂Gbe a connec ed Lie subg oup. Then all o ally geodesic H-o bi s a e cong uen
in M.
P oo . Le H·pand H·qbe wo o ally geodesic o bi s o he ac ion o Hon M. By [4,
P oposi ion 5.5], we ha e ha H·pand H·qha e he same dimension.
Le us suppose ha H·pis a poin . Then, by he homogenei y o M, we ha e
ha H·pand H·qa e cong uen .
Now, le us assume ha dim(H·p)>0. By [4, P oposi ion 5.5] he e exis s an
H-in a ian o ally geodesic submani old No Misome ic o a Riemannian p oduc
N=H·p×Rsuch ha H·p, H·q⊂N. Mo eo e , le γbe he geodesic o Ns a ing
a p∈H·pwhose ini ial eloci y ˙γ(0) lies in he o hogonal complemen o Tp(H·p)
in TpN. This is also a geodesic in Msince Nis o ally geodesic. Fu he mo e,
γhi s H·qa some poin q′∈M, since exp: ν(H·p)→Mis a di eomo phism,
see [115, P oposi ion 3.5 i)]. We may assume q′=γ(1). Le x=γ(1
2)∈N. Then
sx, he geodesic e lec ion o Ma x, maps p∈N o q′∈H·qand p ese es N
since Nis a o ally geodesic submani old in Mcon aining x. Also, i s di e en ial
sends ∈Tp(H·p) o sx∗ ∈Tq′Nand
0 = ⟨ , ˙γ(0)⟩=⟨sx∗ , sx∗˙γ(0)⟩=−⟨sx∗ , ˙γ(1)⟩.
This implies ha sx(H·p) is a o ally geodesic submani old o Npassing h ough
q′∈Nand o hogonal o γ. The angen space o any o bi o His gene a ed by
Killing ec o ields induced by H. Le Xbe a Killing ec o ield induced by he ac ion
o H. Then ∇Xis skew-symme ic, whe e ∇s ands o he Le i-Ci i a connec ion
o N( ecall ha Nis H-in a ian ). Thus ⟨∇˙γX, ˙γ⟩= 0, which implies ha ⟨X, ˙γ⟩is
cons an . Since γis o hogonal o H·pa p, i ollows ha γis also o hogonal o
H·qa q′. Now we ha e shown ha sx(H·p) and H·q, which bo h ha e codimension
one in N, a e o ally geodesic submani olds o Npassing h ough q′wi h he same
angen space. This shows ha H·pand H·qa e cong uen in M ia sx.
7.3 Maximal semisimple o ally geodesic submani olds 127
Rema k 7.3.2.No ice ha he abo e esul wo ks only o symme ic spaces o non-
compac ype. Fo ins ance, he s anda d ac ion o SO2on S2has wo di e en
isome y classes o o ally geodesic o bi s, namely, he equa o and he poles.
Le M=G/Kbe an i educible symme ic space o non-compac ype, whe e G
is he connec ed componen o he iden i y o he isome y g oup o Mand Kis he
iso opy subg oup Goa o∈M. Le Σ be a (comple e) o ally geodesic submani old
o Mpassing h ough o∈M.
As i can be deduced om he discussion in [22,
§
2], i Σ is semisimple, hen he
Lie algeb a gΣ:= [pΣ,pΣ]⊕pΣ, whe e pΣ=ToΣ, is isomo phic o he Lie algeb a
o he isome y g oup o Σ, and gΣis a semisimple Lie algeb a in his case. Recall
ha we say ha a semisimple Lie algeb a is o non-compac ype i each o i s simple
ideals is non-compac . I his a Lie algeb a o non-compac ype, hen he e is some
symme ic space Σho non-compac ype such ha he Lie algeb a o i s isome y
g oup is h. In pa icula , i we conside a Ca an decomposi ion h=kh⊕ph, we ha e
kh= [ph,ph].
Ou ul ima e aim would be o o classi y maximal o ally geodesic submani olds
in M. No e ha Be nd and Olmos classi ied in [20] he maximal o ally geodesic
submani olds o M ha a e non-semisimple. So i emains o ind hose maximal
o ally geodesic submani olds o M ha a e semisimple.
Ou app oach will consis in classi ying i s maximal semisimple o ally geodesic
submani olds (i.e. he o ally geodesic submani olds ha a e maximal among he
semisimple ones) and hen disca ding hose ha a e con ained in a non-semisimple
o ally geodesic submani old. As announced, we will be able o ca y ou hese asks
o he excep ional symme ic spaces, al hough many o he esul s in his chap e
hold in mo e gene ali y.
We now p o e a heo em ha es ablishes a co espondence be ween maximal
semisimple o ally geodesic submani olds o Mand subalgeb as ha a e maximal
among subalgeb as o non-compac ype o g.
Theo em 7.3.3 (Co espondence Theo em).Le M=G/Kbe an i educible sym-
me ic space o non-compac ype and h⊂ga subalgeb a ha is maximal among he
subalgeb as o non-compac ype o g. Then he e is some p∈Msuch ha Σ = H·p
is a maximal semisimple o ally geodesic submani old o M, whe e H⊂Gis he
connec ed subg oup o Gwi h Lie algeb a h.
Con e sely, i Σis a maximal semisimple o ally geodesic submani old o M, hen
he e is a subalgeb a gΣo g ha is maximal among subalgeb as o non-compac ype
o gsuch ha GΣ·p= Σ o some p∈M, whe e GΣis he connec ed subg oup o G
wi h Lie algeb a gΣ.
P oo . Le hbe a maximal subalgeb a o non-compac ype o gand Hbe he con-
nec ed Lie subg oup o Gwi h Lie algeb a h. By Theo em 7.1.2, we may assume ha h
is canonically embedded wi h espec o he Ca an decomposi ion g=k⊕p. Clea ly,
Σ := H·ois hen a semisimple o ally geodesic submani old o Mand we claim ha i
is maximal among semisimple o ally geodesic submani olds. Le us assume ha he e
is a semisimple o ally geodesic submani old e
Σ such ha Σ ⊂e
Σ⊂M. We may assume

128 7 To ally geodesic submani olds in excep ional symme ic spaces
o∈e
Σ. Then e
Σ is o non-compac ype. Le h=kh⊕phbe he Ca an decomposi ion
o h, whe e kh=k∩hand ph=p∩h. I ollows om ou assump ion ha he e is a
Lie iple sys em pe
Σsuch ha ph⊂pe
Σ⊂p. Thus, kh= [ph,ph]⊂[pe
Σ,pe
Σ] =: ke
Σand
h⊂ge
Σ:= ke
Σ⊕pe
Σ. Howe e , ge
Σis o non-compac ype, since e
Σ is o non-compac
ype, and his maximal among subalgeb as o go non-compac ype. I ollows ha
h=ge
Σand e
Σ = Σ.
Le Σ be a maximal semisimple o ally geodesic submani old passing h ough
o∈Mand le pΣbe he angen space o Σ a o. Conside gΣ:= [pΣ,pΣ]⊕pΣ. Clea ly,
gΣis o non-compac ype since i is isomo phic o he Lie algeb a o he isome y
g oup o Σ. We will p o e ha gΣis maximal among subalgeb as o non-compac
ype o g. Le hbe a subalgeb a o go non-compac ype such ha gΣ⊂h⊂g. We
p o e ha gΣ=h. We apply Theo em 7.1.2 o gΣ⊂h o ind a Ca an decomposi ion
o hsuch ha
h=kh⊕phsa is ying kΣ:= [pΣ,pΣ]⊂kh,pΣ⊂ph.
Applying Theo em 7.1.2 once mo e o h⊂g, we ind a Ca an decomposi ion o g
such ha
g=k′⊕p′sa is ying kΣ⊂kh⊂k′,pΣ⊂ph⊂p′.
Now since any wo Ca an in olu ions in a eal semisimple Lie algeb a di e by an
inne au omo phism, he e is some g∈Gsuch ha p′= Ad(g)p. Thus Ad(g−1)pΣ
and Ad(g−1)pha e Lie iple sys ems in psince Ad(g)∈Au (g) and Ad(g−1)pΣ⊂
Ad(g−1)ph⊂p. Le Hand GΣbe he Lie subg oups o Gwi h Lie algeb as hand
gΣ, espec i ely. I we conside p:= g·o∈M, we ha e ha GΣ·pand H·pa e
o ally geodesic submani olds in M, since g−1GΣg·oand g−1Hg·oa e o ally geodesic
(as Ad(g−1)pΣand Ad(g−1)pha e Lie iple sys ems in p). Fu he mo e, GΣ·pis
con ained in H·p. Howe e , Σ = GΣ·ois maximal among he semisimple o ally
geodesic submani olds passing h ough o∈M. Hence, by P oposi ion 7.3.1, GΣ·pis
maximal among he semisimple o ally geodesic submani olds passing h ough p∈M.
Then GΣ·p=H·p. The e o e, ph=pΣand consequen ly gΣ=h, since gΣand ha e
o non-compac ype.
7.4 Dynkin index and o ally geodesic submani olds
In his sec ion, we ex end he de ini ion o he Dynkin index o a simple subalgeb a
o a simple complex Lie algeb a o ce ain classes o semisimple subalgeb as o simple
eal Lie algeb as, and we use i o cha ac e ize isome y classes o o ally geodesic
submani olds.
Recall ha we deno e by Bg he Killing o m o a Lie algeb a g. Le gbe a simple
complex Lie algeb a and le abe a Ca an subalgeb a o g. Le ∆ be he se o oo s
wi h espec o a. I is shown in [90, Chap e III, Theo em 4.2] ha he es ic ion
o Bg o ais non-degene a e, and ha we hence may iden i y each oo α∈∆ wi h
a ec o Hα∈gsuch ha α(H) = Bg(H, Hα) o all H∈a. I ollows om [90,
7.4 Dynkin index and o ally geodesic submani olds 129
Chap e III, Theo em 4.4(i)] ha he numbe s qα:= Bg(Hα, Hα) a e posi i e o
all α∈∆. Le q:= max{qα:α∈∆}and de ine he bilinea o m Qgon gby
Qg:= 2
qBg,
i.e. Qgis he mul iple o he Killing o m no malized so ha he squa e o he leng h
o he longes oo equals 2. Le us ecall om [74,
§
2] Dynkin’s de ini ion o he
index o a simple subalgeb a o a simple complex Lie algeb a. Le hand gbe simple
complex Lie algeb as and le :h→gbe a Lie algeb a monomo phism. By Schu ’s
Lemma, he numbe indD( ), gi en by
indD( )·Qh(X, Y ) = Qg( (X), (Y)) o all X, Y ∈h,(7.1)
is well de ined. I is called he Dynkin index o he subalgeb a (h) in g. Dynkin
p o ed in [74, Theo em 2.2] ha i is a posi i e in ege . I h⊂gis a complex
subalgeb a o he complex Lie algeb a g, we will w i e indD(h,g), o indD(h) when
he embedding is clea om he con ex , o he Dynkin index o he inclusion map.
Now le us men ion he mul iplica i e p ope y o he Dynkin index, see [74,
§
2,
No. 7]. Obse e ha gi en he inclusions h1⊂h2⊂go complex simple Lie algeb as,
hen
indD(h1,g) = indD(h1,h2)·indD(h2,g).
We would like o ex end he de ini ion o he Dynkin index o (semi)simple sub-
algeb as o simple eal Lie algeb as.
Le h=h1⊕···⊕hnbe a semisimple complex subalgeb a o he simple complex
Lie algeb a g, whe e hsis a simple complex ideal o e e y s∈ {1, . . . , n}. We de ine
indD(h) := (indD(h1), . . . , indD(hn)).
Fo an absolu ely simple eal Lie algeb a g, we de ine he Dynkin index o a
semisimple subalgeb a has he Dynkin index o hCin gC. This is well de ined since
hCis semisimple, see he p oo o Lemma 7.2.3. In o de o de ine also he Dynkin
index o a semisimple subalgeb a o a simple eal Lie algeb a ha is no absolu ely
simple, i.e. whose complexi ica ion is no simple, we p o e he ollowing.
Lemma 7.4.1. Le gbe a simple complex Lie algeb a and le hbe a semisimple
subalgeb a o he eali ica ion gRo g. Then h+ihis a semisimple subalgeb a o g.
P oo . No ice ha h+ihis a subalgeb a o gand h∩ihis an ideal o h+ih, see [73,
Appendix o Chap e 1, Lemma 1.1]. Hence h∩ihis also an ideal o he subalgeb a
ho gR. Le h=h1⊕ ···⊕hn, whe e he hja e he simple ideals o h. Since hjis
simple, we ha e ei he hj∩ihj=hj, and hen hj+ihj=hjis simple, o h∩ih= 0, in
which case hjis a simple eal o m o he complex Lie algeb a hj+ihj. No e also ha
ihj∩hk= 0 o j=k. Indeed, we ha e [ihj∩hk,hk]⊂i[hj,hk] = 0 o j=k. Thus
we may assume, by enumbe ing he hj, i necessa y, ha he e is a k∈ {0, . . . , n}
such ha mul iplica ion by imaps hj o i sel i and only i j < k. I ollows ha
h+ih=h1⊕···⊕hk−1⊕(hk+ihk)⊕···⊕(hn+ihn).
Hence h+ihis semisimple.
130 7 To ally geodesic submani olds in excep ional symme ic spaces
De ini ion 7.4.2. Le gbe a simple eal Lie algeb a and le hbe a semisimple sub-
algeb a o g.
i) I gis absolu ely simple, hen we de ine
indD(h,g) := indD(hC,gC).
ii) I g=lRis he eali ica ion o a simple complex Lie algeb a land his a complex
subalgeb a o a eal o m o l, hen we de ine
indD(h,g) := indD(h+ih,l).
Fu he mo e, le hand h′be wo semisimple subalgeb as o g o which he Dynkin
index is de ined. We say ha hand h′a e isome ic i he e is a Lie algeb a isomo -
phism φ:h→h′such ha each simple ideal o his mapped on o a simple ideal o h′
o he same Dynkin index.
Rema k 7.4.3.No e ha , wi h his de ini ion, o a simple subalgeb a ho an abso-
lu ely simple eal Lie algeb a g he Dynkin index o he subalgeb a hCo gCis ei he
a na u al numbe (in case his absolu ely simple) o , by he ollowing lemma, a pai
o na u al numbe s (in case his no absolu ely simple).
Lemma 7.4.4. Le gbe an absolu ely simple eal Lie algeb a and le hbe a simple,
bu no absolu ely simple subalgeb a o g. Then hC=h1⊕h2, whe e h1and h2a e
wo isomo phic simple subalgeb as o gCo equal Dynkin index.
P oo . We may assume ha his canonically embedded wi h espec o a Ca an
decomposi ion g=k⊕pby Theo em 7.1.2. Le θbe he co esponding Ca an in olu-
ion. Since his simple, bu no absolu ely simple, i is isomo phic o he eali ica ion
o a simple complex Lie algeb a and since his canonically embedded wi h espec o
he abo e Ca an decomposi ion, i ollows ha h∩kis a compac eal o m o his
simple complex Lie algeb a. In pa icula , θ, es ic ed o h, is a non- i ial in olu i e
au omo phism o h.
I is well known ha he complexi ica ion o R, whe e is a complex Lie algeb a,
is isomo phic o ⊕¯
, whe e ¯
deno es he complex conjuga e Lie algeb a o , see
e.g. [150,
§
2, P oposi ion 3]. No e ha he complex Lie algeb as ha ha e a eal o m
a e isomo phic o hei complex conjuga es ia an an ilinea map. This shows ha
hC=h1⊕h2, whe e h1and h2a e isomo phic.
Le τ:gC→gCbe he map de ined by τ(X+iY ) = X−iY o X, Y ∈g. I
is s aigh o wa d o check ha τis an au omo phism o gR. The map τob iously
lea es hCin a ian and i ac s on hCas an au omo phism o he eal Lie algeb a (hC)R.
Since h1and h2a e isomo phic simple ideals o (hC)R, we ha e ei he τ(h1) = h1o
τ(h1) = h2. Assume we a e in he o me case. Then we also ha e τ(h2) = h2. The
ixed poin se o he in olu ion τon hCis he di ec sum k1⊕k2, whe e kjis a p ope
subalgeb a o hj o j= 1,2. Howe e , he ixed poin se o he ac ion o τon hC
coincides wi h he simple Lie algeb a hand we ha e a i ed a a con adic ion.
7.4 Dynkin index and o ally geodesic submani olds 131
We ha e shown ha τ(h1) = h2. Since h1is simple, he e a e ec o s X, Y ∈h1
such ha Bh1(X, Y )= 0 and we ha e by [150,
§
2, P oposi ion 2(ii)]
indD(h1,g) = Qg(X, Y )
Qh1(X, Y )=Qg(τ(X), τ(Y))
Qh2(τ(X), τ(Y)) = indD(h2,g).
Since indD(h2,g) is a na u al numbe , i ollows ha he subalgeb as h1and h2ha e
he same Dynkin index.
Hence, we ha e de ined he Dynkin index o e e y semisimple subalgeb a ho an
absolu ely simple eal Lie algeb a g. Also, we ha e de ined he Dynkin index o e e y
semisimple complex subalgeb a ho he eali ica ion o a complex simple Lie algeb a
gand o i s eal o ms. In bo h cases we deno e i by indD(h,g), o indD(h), when
he embedding is clea om he con ex . I immedia ely ollows om his de ini ion
ha he Dynkin index is one i his a eal o m o he simple complex Lie algeb a g.
Indeed, he ollowing esul shows ha isome ic eal o ms o ginduce cong uen ,
and hence isome ic, o ally geodesic submani olds o M.
Lemma 7.4.5. Le M=G/Kbe a symme ic space, whe e Gis a simple complex
Lie g oup. Le hand h′be isomo phic eal o ms o a complex simple Lie algeb a g.
Then, he o ally geodesic o bi s o Hand H′in Ma e all cong uen in M, whe e H
and H′a e he connec ed subg oups o Gwi h Lie algeb as hand h′, espec i ely.
P oo . Le hand h′be isomo phic eal o ms in g. Then, he e is a Lie algeb a isomo -
phism :h→h′. By complexi ying, his map ex ends o a Lie algeb a au omo phism
e
o g. Le us ix some Ca an decomposi ion g=k⊕p. We can assume wi hou loss
o gene ali y ha H·ois a o ally geodesic submani old in M. This implies ha he e
is a Ca an decomposi ion h=ph⊕khsuch ha ph⊂pand kh⊂k. By Theo em 7.1.2,
he e is a Ca an decomposi ion ph′⊕kh′=h′such ha e
(ph) = ph′and e
(kh) = kh′.
Fu he mo e, wo Ca an decomposi ions o ga e conjuga e in In (g). Hence he e
is some g∈Gsuch ha φ:= Ad(g)◦e
∈Au (g), φ(p) = pand φ(h)⊂gis a eal
o m o gconjuga e o h′in g. Fu he mo e, φp ese es he cu a u e enso o
Ma osince his is gi en by Lie b acke s. Hence, φis a linea isome y o p ha
p ese es sec ional cu a u e a o. Thus, by [189, Co olla y 2.3.14], φex ends o an
isome y k∈Isom(M) ha ixes o∈M, since i lea es pin a ian . Thus, we ha e
ha k(H·o) = gH′g−1·q, o ce ain q∈M, which implies ha he e is a o ally
geodesic o bi o Hwhich is cong uen o a o ally geodesic o bi o H′. Consequen ly,
by P oposi ion 7.3.1, he o ally geodesic o bi s o Hand H′a e all cong uen in
M.
We ha e de ined he Dynkin indices o ce ain semisimple subalgeb as o simple
eal Lie algeb as in such a way ha isome y classes o subalgeb as o he isome y
algeb a o an i educible symme ic space o non-compac ype co espond o isome y
classes o o ally geodesic submani olds. This is he con en o he ollowing heo em.
Theo em 7.4.6. Le M=G/Kbe an i educible symme ic space o non-compac
ype. Le Σ1,Σ2be wo semisimple o ally geodesic submani olds con aining o∈M.