Locally
homogeneous
aspherical
Sasaki
manifolds
Oliver Bauesa,∗, Yoshinobu Kamishimab
aDepartmentofMathematics,UniversityofFribourg,CheminduMusée23,CH-1700Fribourg,
Switzerland
bDepartmentofMathematics,JosaiUniversity,Keyaki-dai1-1,Sakado,Saitama350-0295,Japan
a b s t r a c t MSC: 57S30 53C12 53C25 Keywords:
LocallyhomogeneousSasaki manifold
Asphericalmanifold Pseudo-Hermitianstructure CR-structure
LocallyhomogeneousKähler manifold
Let G/H be a contractible homogeneous Sasaki manifold. A compact locally homogeneous aspherical Sasaki manifold ΓG/H is by definition a quotient of
G/H by a discreteuniform subgroup Γ ≤ G. We show that a compact locally homogeneousaspherical Sasakimanifold is alwaysquasi-regular, thatis, ΓG/H
isanS1-SeifertbundleoveralocallyhomogeneousasphericalKählerorbifold.We discussthestructureoftheisometrygroupIsom (G/H) foraSasakimetricofG/H inrelation withthe pseudo-Hermitiangroup Psh (G/H) for theSasaki structure of G/H. We show that a Sasaki Lie group G, when ΓG is a compact locally homogeneousaspherical Sasakimanifold,iseithertheuniversalcoveringgroupof SL(2,R) ora modification of a Heisenberg nilpotent Lie group with itsnatural Sasakistructure.WealsoshowthatanycompactregularasphericalSasakimanifold withsolvablefundamentalgroupisfinitelycoveredbyaHeisenbergmanifoldand itsSasakistructuremaybedeformedtoalocallyhomogeneousone.Inaddition,we classifyallasphericalSasakihomogeneousspacesforsemisimpleLiegroups.
1. Introduction
LetM beasmoothcontactmanifoldwithcontactformω.Supposethatthereexistsacomplexstructure
J on thecontact bundle ker ω and thatthe Levi formdω◦ J is apositivedefinite Hermitian form.Then
{ω,J} iscalledapseudo-Hermitianstructure onM and{ker ω,J} isaCR-structureaswell.Thepair{ω,J}
assignsaRiemannianmetricg toM ,where
g = ω· ω + dω ◦ J. (1.1)
Therearetwotypical, closelyrelated,Liegroupson(M,{ω,J}).Thegroupofpseudo-Hermitian transfor-mationsofM is denotedby
* Correspondingauthor.
E-mailaddresses:[email protected](O. Baues),[email protected](Y. Kamishima).
http://doc.rero.ch
Published in "Differential Geometry and its Applications 70(): 101607, 2020"
which should be cited to refer to this work.
Psh (M ) ={h ∈ Diff (M) | h∗ω = ω, h∗◦ J = J ◦ h∗ on ker ω}. AsusualIsom (M ) denotestheisometrygroupof(M,g).Obviously
Psh (M )≤ Isom (M) .
AssumethattheReebfieldA forω generatesaone-parametergroupT ofholomorphictransformationson aCR-manifold(M,{ker ω,J}),thatis,
T ≤ Psh (M) .
Then (M,{ω,J}) issaid to beastandard pseudo-Hermitian manifold.In thiscase, thevectorfieldA is a Killing fieldof unit lengthwith respect to g, and the Riemannianmanifold (M,g) is alsocalled aSasaki manifold equippedwithSasaki metricg andstructurefield A.IfA isacompletevectorfieldwithaglobal flowT whichactsfreelyandproperlyonM , (M,{g,A}) issaidto bearegularSasaki manifold.Notethat the Sasaki metric structure (M,{g,A}) determines the standard pseudo-Hermitian structure (M,{ω,J})
uniquely.
The pseudo-Hermitian group Psh (M ) and isometry group Isom (M ) of a Sasaki manifold are closely related.SincetheReebvectorfieldA isdeterminedbyω alone,wehave
h∗A = A , for all h ∈ Psh (M) . Therefore,theReebflowT belongstothecenterof Psh (M ),thatis,
Psh (M ) = CPsh (M)(T ) . (1.2)
Similarly,ifCIsom (M)(T ) denotesthecentralizer ofT inIsom (M ),using(1.1), Psh (M ) = CIsom (M)(T )
followseasily,as well.
Ingeneral,thegroupIsom (M ) actsontheset ofSasaki structures{g,A} withfixedmetricg. Further-more, if(M,g) is not isometrically coveredby around sphere,the set of Sasaki structures withmetric g
eitherconsistsoftwoelements{A,−A},orM isathree-Sasakimanifold,admittingthreelinearindependent Sasaki structuresfor g.Inthe lattercase,M iscompact withfinite fundamentalgroup.For these results, see[26,20,27].Thus,unlessM iscompactwithfinitefundamentalgroup,acompleteSasakimanifoldalways satisfies
Isom (M ) = Psh±(M ) ={h ∈ Isom (M) | h∗A = ±A} .
CallaSasakimanifoldM ahomogeneousSasakimanifold ifPsh (M ) actstransitivelyonM .Accordingly, ahomogeneousspace G/H iscalled ahomogeneous Sasaki manifold if G/H isaSasaki manifold and the action of G factors over Psh (G/H).Note thatany homogeneous Sasaki manifold is alsoa regularSasaki manifold.
1.1. LocallyhomogeneousasphericalSasaki manifolds
Inthefollowing we shallusually assumethatG acts effectivelyonG/H andthereby identifyG with a closedsubgroupof Psh (G/H) wheneversuitable.
A locallyhomogeneous Sasakimanifoldis aquotientspace
M = ΓG/H
ofahomogeneousSasakimanifoldG/H byadiscretesubgroupΓ ofG.ThemanifoldM iscalledaspherical
ifitsuniversalcoverX iscontractible.Inthispaperwetakeupthestructureofcompactlocallyhomogeneous aspherical SasakimanifoldsM .
Settingthestage forthemain structureresultoncompactlocallyhomogeneous aspherical Sasaki mani-folds,wenote thefollowing facts:
Let X = G/H be ahomogeneous Sasaki manifold. Then the Reeb flow T on X is isomorphic to the real line R or thecircle group S1 and itis acting freely and properly on X. Moreover, thehomogeneous
pseudo-HermitianstructureonX inducesauniquehomogeneous Kählerstructureonthequotientmanifold
W = X/T
such thatthe projection map X → W is a principal bundle projection which is pseudo-Hermitian (that is,X → W ishorizontallyholomorphic andhorizontallyisometric). Withthis structure thehomogeneous KählermanifoldW willbe calledtheKähler quotient ofX.
ThereisawellestablishedtheoryofhomogeneousKählermanifoldsandofthestructureoftheirisometry groups[12].As acornerstoneof thepresent paperwe usethis theoryto develop thestructure ofcompact locally homogeneous Sasaki manifolds. Recall that a Lie group is called unimodular if its Haar measure is biinvariant. Any Lie group G which admits a uniform lattice Γ is unimodular. Thus, in our context, homogeneousSasaki spaces andKähler manifolds of unimodularLie groupsare ofparticular importance. Thestructuretheory ofsuch spacesalreadyplays amajor roleinthepapers[1,16]. Asastartingpoint of thepresentworkwe refineandalso detailtheapproachthereinto study contractiblehomogeneousSasaki spacesofunimodularLiegroupsandtheircorrespondingKähler quotients.
1.1.1. Structure oflocallyhomogeneousasphericalSasaki manifolds
Let us first introduce some additional notation. For any Kähler manifold W , we denote Isom±h(W ) the subgroup of Isom (W ) that consists of isometries which are either holomorphic or anti-holomorphic. Furthermore,Isomh(W ) denotesthesubgroupofholomorphic(orKähler-)isometries ofW .
ThemainstructureresultonlocallyhomogeneousasphericalSasakimanifoldsandtheirisometrygroups isstatedinthefollowingtwo results:
Theorem 1.Let X = G/H be a contractible homogeneous Sasaki manifold of a unimodular Lie group G. Thenthefollowinghold:
(1) The KählerquotientW ofX is aproductofaunitary spaceCk witha boundedsymmetricdomainD.
(2) The Reeb flow T is isomorphic to thereal lineand it isa normal subgroup of Isom (X).There exists an inducedhomomorphism
φ : Isom (X)→ Isom±h(W ) ,
whichisonto andmapsPsh (X) onto Isomh(W ) with kernelT .
(3) Thereexistsan antipseudo-Hermitian involutionτ ofX such that
Isom (G/H) = Psh±(G/H) = Psh (G/H) τ .
(4) Theidentity component of thepseudo-Hermitiangroupof X satisfies
Psh (G/H)0= Isom (G/H)0= (N U(k)) · S ,
where N is a 2k + 1-dimensional Heisenberg Lie group and S is a normal semisimple Lie subgroup whichcoverstheidentity component
S0= Isom (D)0
of the isometry group of the symmetricbounded domain D. Moreover, S has infinite cyclic centerΛ,
and
S∩ N = S ∩ T = Λ .
BuildingonTheorem 1wecan deduce:
Corollary1.LetM = ΓG/H beacompactlocallyhomogeneousasphericalSasakimanifold.Thenthecoset spaceΓG/H admitsan S1-bundle overalocallyhomogeneousasphericalKähler orbifold
S1 −−−−→ ΓG/H −−−−→ φ(Γ)W , (1.3)
inwhich S1 inducestheReebfield. In particular,theSasaki manifoldM isquasi-regular.
Remark 1.1. The bundle in (1.3) is called a Seifert fibering. Here, some finite covering space Γ0
G/H,
withΓ0≤ Γ afiniteindex subgroup,is anon-trivialS1-bundleoveraKählermanifold φ(Γ0)
W .Note, in addition,thatforanySasakimanifoldM = ΓG/H asabove,Psh (ΓG/H)0containstheflowoftheReeb field.Thisflowisacompactone-parametergroupS1actingalmostfreelyonM anditisgivingriseto the
bundle(1.3).Moreover,sincetheSasakistructureonM arisesfromaconnectionform,theKählerclassof
φ(Γ0)
W representsthecharacteristicclass ofthecirclebundle. Wefurtherremark:
(5) When the anti-holomorphic isometry τ ofX from Theorem 1normalizes Γ,we getIsom (ΓG/H)= Psh (ΓG/H) Z2, otherwisewehaveIsom (Γ
G/H)= Psh (ΓG/H).
LetN denotethe2n+ 1-dimensional Heisenberggroupwith itsnaturalSasakimetric.Using (5)above wealsoget:
(6) ThereexistsacompactlocallyhomogeneousasphericalRiemannianmanifold
M = π\N ,
whosemetricislocallyaSasakimetric(thatis,itisinducedfromtheleft-invariantSasakimetriconN ). ButM with metricg isnotaSasaki manifolditself.
1.1.2. The caseof solvablefundamental group
Wesuppose thatthefundamentalgroupofthe compactasphericalmanifold M is virtuallysolvable.In thiscase,ifM supportsalocallyhomogeneousSasakistructure,thenTheorem1impliesthatM isfinitely coveredbyaHeisenbergmanifold
Δ\ N ,
whereΔ≤ N isauniformdiscretesubgroupofN .Moreover,M isanon-trivialcirclebundleoveracompact flatKählermanifold, which inturnis finitely coveredby acomplextorusCk/Λ.As amatter offact, any
compactaspherical Kähler manifoldis biholomorphicto aflatKähler manifold (see [4, Theorem 0.2] and thereferencestherein).Asaconsequence,anyregularSasakimanifold M isoftheabovetypeaswell,and itadmitsalocally homogeneousSasakistructure:
Corollary 2.Let M be a regular compact aspherical Sasaki manifold with virtually solvable fundamental group.Then thefollowinghold:
(1) The manifold M is a circle bundle over a Kähler manifold that is biholomorphic to a flat Kähler manifold.
(2) A finitecover ofM is diffeomorphic toaHeisenbergmanifold.
Moreover,theSasakistructureonM canbedeformed(viaregularSasakistructures)toalocallyhomogeneous Sasakistructure.
1.2. ContractibleSasaki Liegroups andcompactquotients
We callaLiegroupG a Sasaki group ifitadmits aleft-invariant Sasakistructure.Equivalently,G acts
simplytransitivelybypseudo-Hermitian transformationsonaSasaki manifoldX.
A prominentexampleofaSasaki Liegroupisthe2n+ 1-dimensionalHeisenbergLie groupN .TheLie groupN arises asanon-trivialcentralextensionoftheform
R → N → Cn ,
andanaturalSasaki structureonN isobtainedbyaleft-invariantconnectionformwhichisassociated to thiscentralextension.
More generally,weshall introduceafamilyof simplyconnected2n+ 1-dimensionalsolvable Sasaki Lie groups
N (k, l) , k + l = n,
called Heisenberg modifications. These groups are deformations of N in N Tk, where Tk ≤ U(n) is a compacttorus (cf.Definition7.7).
AnothernoteworthycontractibleLie groupwhichisSasakiis
SL(2,R) ,
theuniversalcoveringgroupofSL(2,R).Indeed,takeany left-invariantmetricg onSL(2,R) withthe addi-tionalpropertythatg isalsoright-invariantbytheone-parametersubgroupSO(2,R).ThentheRiemannian submersionmap
SL(2,R) → SL(2,R) SO(2,R) = HC1
isdefinedanditisaprincipalbundlewithgroupSO(2,R) = R overaRiemannianhomogeneousspaceH1 Cof
constantnegativecurvature.Themetricg definesauniqueleft-invariantconnectionformω,whichsatisfies
(1.1) andhastheproperty thattheReebfieldisleft-invariantand tangentto thesubgroupSO(2,R). The isomorphismclassesof Sasakistructuresthusobtainedareparametrizedbythecurvatureofthebase.
In general, a simply connected unimodular Lie group that admits a left-invariant Sasaki structure is isomorphic to either of the Lie groups SU(2), N or the universal covering group of SL(2,R), see [16, Theorem4] andalso[8,Theorem5].(In[1,Theorem2.1] amoreprecisestatementisobtainedbydescribing left-invariantSasaki structuresuptoamodification process.)
Asanapplicationof ourmethodsweexplicitly statetheclassificationof contractibleunimodularSasaki Liegroupsasfollows:
Theorem2.LetG beaunimodularcontractibleSasakiLiegroup.ThenasaSasakiLiegroupG isisomorphic toeither N (k,l) orSL(2,R) with oneof theleft invariantSasaki structures asintroduced above. (Thatis, G admitsapseudo-Hermitian isomorphismtoeitherN (k,l) orSL(2,R) withastandardSasaki structure.)
Remark1.2.AsintroducedabovethefamilyofallSasakiLiegroupsN (k,l) isinonetoonecorrespondence with the set of isomorphismclasses of flatKähler Lie groups.Compare Section7.2.2. For adiscussion of thestructureofflatKählerLiegroups,seeforexample[13] or[3].
Remark1.3.Whendroppingtheassumptionofcontractibility,thecompactgroupSU(2) appearsasanother unimodularSasakiLie group.Thisgroupisfiberingovertheprojective lineP1C,and theexampleisdual
to theSasaki Lie groupSL(2,R). The twogroups areknownto be the onlysimply connected semisimple Liegroupswhichadmitaleft-invariant Sasakistructure,cf.[8,Theorem5].
AnyLiegroupG whichadmitsadiscreteuniformsubgroupΔ mustbeunimodular,andifsuchG admits
thestructureofaSasakiLiegroupthenthequotientmanifold ΔG
inheritsthestructure ofacompactlocally homogeneousSasaki manifold. Thus,combiningTheorem 2withCorollary 1weobtain:
Corollary3.EverycompactlocallyhomogeneousasphericalSasaki manifoldwhichisof theform
Δ\G
iseitheraSeifertmanifold,whichisanS1-bundleoverahyperbolictwo-orbifold,oritisaSeifertmanifold whichisan S1-bundleover aflatKähler manifold(whichisacomplex torusbundleoveracomplex torus). 1.3. Sasaki homogeneousspacesof semisimple Liegroups
HereweconsiderthequestionwhichsemisimpleLiegroupsacttransitivelybypseudo-Hermitian transfor-mationsonacontractible(or,moregenerally,aspherical)Sasakimanifold.Theclassificationofsuchgroups andofthecorrespondinghomogeneousspacesiscontainedinTheorem3followingbelow.
LetD be abounded symmetricdomain,equipped with itsnaturalBergman Riemannianmetric. Then itsisometrygroup
S0= Isom (D)0
isasemisimpleLiegroupwhichiscalled agroupofHermitiantype,and
D = S0/K0
isaRiemanniansymmetricspacewithrespecttothismetric,andahomogeneousKählermanifold,aswell. Moreover,K0 isamaximalcompactsubgroupofS0.
Theorem3.Forany symmetricboundeddomainD = S0/K0,thereexistsauniquesemisimple LiegroupS withinfinite cyclic center,which iscoveringS0,andgives risetoacontractibleSasaki homogeneousspace
XS= S/K
withKählerquotientD.Moreover,anycontractiblehomogeneousSasakimanifoldofasemisimpleLiegroup isofthis type.
Addendum: Inthetheorem,K isamaximalcompactsubgroupofS,andS≤ Psh (XS) isactingfaithfully onXS.TheKählerquotientD isahomogeneousKählermanifoldwhosecomplexstructureisbiholomorphic to a bounded symmetric domain. It carries an invariant symmetric Kähler Riemannian metric, which is uniqueuptoscaling onirreducible factorsofthehomogeneousspaceD.
As aconsequenceofTheorem 3anyLiegroupofHermitiantypeactsasatransitivegroupofisometries onanasphericalSasaki space:
Corollary4. Forany semisimpleLiegroupS0 ofHermitiantype,thereexistsauniqueSasakihomogeneous space
Y = S0/K1,
whereY isacircle bundleover thesymmetricboundeddomainD = S0/K0.
Note that,inTheorem3andCorollary4theSasaki structureonX,respectivelyY ,isuniqueuptothe choiceofanS0-invariant(andalsosymmetric)KählermetriconD.
Thepaperisorganizedasfollows. StartinginSection2,wecollectandexplainsomeusefulbasicfactson regularSasakimanifolds,includingtheBoothby-Wangfibrationandthejoinconstruction.
In Section 3 we discuss the lifting of Kähler isometries and the role of gauge transformations in the Boothby-WangfibrationofacontractibleSasaki manifold.
We use these facts to show thatevery contractible homogeneousKähler manifold determines aunique
contractiblehomogeneousSasakimanifold.AlsotheassociatedpresentationsofahomogeneousSasaki man-ifoldbytransitivegroupsof pseudo-Hermitiantransformationarediscussed inSection4.
Section5isdevotedtothestudyofhomogeneouscontractibleKählermanifoldsofunimodularLiegroups. Theirclassification is derivedfrom theDorfmeister-Nakajima fundamentalholomorphic fiber bundle of a homogeneousKähler manifold.
ThestructureoflocallyhomogeneousasphericalSasakimanifoldsispickedupinSection6.Weestablish inCorollary1thatacompactlocally homogeneousaspherical Sasakimanifoldisalwaysquasi-regularover acompact orbifoldwhichismodeledonahomogeneouscontractible Kählermanifold. Therelevantglobal resultsare summarizedinTheorem 1and itsproof.Wealso givetheproofofCorollary 2inSection6.4.1, seeinparticular Proposition6.10.
In Section7weturn ourinterest to theclassificationproblem forglobal modelspaces oflocally homo-geneousSasaki manifolds:inparticular,we classifycontractibleSasaki Lie groupsandcontractible Sasaki homogeneousspacesofsemisimpleLie groups.Inthecourse,weproveTheorem2and Theorem3.
InSection8weconstructfurtherexplicitexamplesof locallyhomogeneousasphericalSasakimanifolds. Referto [25],[7],[11] forbackgroundonSasaki metricstructures ingeneral.
2. Preliminaries
LetX = (X,{ω,J}) beaSasakimanifoldwithReebflowT . 2.1. Regular Sasakimanifolds
TheSasaki manifold X is called regularif theReeb flowT is complete and T acts freely andproperly onX.Inthis situation,eitherT =R,or T = S1isacirclegroup.Moreover,
W = X/T
isasmoothmanifoldandX isaprincipalbundleoverW withgroupT .
Example2.1. LetX = G/H beahomogeneousSasakimanifold. ThenX isregular.(See [8] andSection4 below.)
Forthefollowing,see[8]:
Proposition2.2(Boothby-Wangfibration). LetX bearegularSasakimanifoldwithReebflow T .Thenthere isan associatedprincipalbundle
T → X→ Wq
overaKähler manifold(W,Ω,J) suchthattheinducedisomorphism q∗: ker ω → T W
isholomorphicand theKähler formon thebaseissatisfying theequation
q∗Ω = dω . (2.1)
Furthermore,there isanaturalinducedhomomorphism
Psh (X)−→ Isomφ h(W ) (2.2)
with kernelT , whichissatisfying q◦ ˜h = φ(˜h)◦ q,forallh˜∈ Psh (X).
Withtheaboveconditionssatisfied,wecall(W,Ω,J) theKählerquotient oftheregularSasakimanifold X. Alsowelet
Isomh(W ) = Isom (W, Ω, J) denotethegroupofholomorphicisometriesoftheKählerquotientW .
Proof of Proposition2.2. The projectionq inducesanisomorphism
q∗ : ker ω−→T W
ateachpoint.Since ω isinvariantunderT ,ω inducesawelldefined2-formΩ onW suchthat
dω(X , Y) = Ω(q∗X , q∗Y) , forallhorizontalvectorfieldsX ,Y ∈ ker ω thatarehorizontallifts.As
ιAdω = 0 ,
itfollowsthatdω = q∗Ω and sodΩ= 0.SincetheReebflow T isholomorphic onker ω,usingJ onker ω,
q∗ inducesawelldefinedalmostcomplexstructureJ onˆ W suchthatΩ isJ-invariant.ˆ SinceJ isintegrable (thatis,[T1,0,T1,0]⊆ T1,0fortheeigenvaluedecompositionker ω⊗ C = T1,0⊕ T0,1),J becomesˆ acomplex structureonW .HenceΩ isaKählerformonthecomplexmanifold(W,J).ˆ Tosimplifynotation,fromnow on,thesamesymbolJ isusedforthecomplexstructureonW ,forwhichwerequirethatq isaholomorphic map onker ω,thatis, theinducedisomorphismq∗: ker ω → T W satisfiesq∗◦ J = J ◦ q∗.
Since it is commutingwith theprincipal bundleaction of T ,which is arising fromtheReeb flow, each holomorphicisometry
˜
h∈ Psh (X) = CPsh (X)(T ) inducesadiffeomorphismh: W → W ,suchthatthediagram
X −−−−→ X˜h
q⏐⏐ q⏐⏐
W −−−−→ Wh
(2.3)
is commutative. (We briefly verify that h∗Ω = Ω and h∗◦ J = J ◦ h∗ on W : Indeed, as ˜h∗dω = dω, it followsby(2.1) thatq∗(h∗Ω)= ˜h∗q∗Ω= q∗Ω.Thisshowsh∗Ω= Ω.Sinceq∗J = J q˜ ∗onker ω,using(2.3) it followsh∗J q∗(Y)= h∗q∗J(Y)˜ = q∗˜h∗J(Y)˜ = q∗J ˜˜h∗(Y) = J h∗q∗(Y), for all vector fields Y ∈ ker ω,which arehorizontal liftsfor avector fieldonW . So h∗◦ J = J ◦ h∗ onW .) Thus h is aholomorphic isometry of W .
Furtheranylift˜h∈ Psh (X) of h isuniqueuptocompositionwith anelement oftheReebflow:Indeed, supposethath= idW.Since T actstransitively onthefibers,after compositionwithanelement of T ,we mayassume thatthere exists afixedpoint x∈ X for ˜h.Moreover, since˜h∗A=A, thedifferentialof ˜h at
x istheidentityofTxX.Noweveryisometry ˜h oftheRiemannianmanifoldX is determinedbyitsone-jet atonepoint x.Hence,ker φ= T .
2.2. Holomorphic andanti-holomorphicisometries
Forany SasakimanifoldX withReebfieldA,webriefly recalltheinteractionof Psh±(X) ={h ∈ Isom (X) | h∗A = ±A}
with the pseudo-Hermitian structure of X. For any pseudo-Hermitian structure {ω,J}, the structure
{−ω,−J} is called theconjugate structure.Then the groupof isometries Psh±(X) permutesthe pseudo-HermitianstructureofX anditsconjugate:
Lemma 2.3 (Sasaki isometries).Let X be any Sasaki manifold and let h∈ Isom (X) satisfy h∗A = ±A,
whereA istheReeb fieldof X.Thenh∗ω =±ω andh∗J =±Jh∗ on ker ω.
Proof. Forany X ∈ ker ω,theequationg(h∗A,h∗X )= g(A,X ) shows 0 = g(X , A) = ω(X )ω(A) + dω(JX , A) = ω(X )
= g(h∗X , h∗A) = ±g(h∗X , A) = ±ω(h∗X ) . Inparticular,h∗ mapsker ω ontoitself.As
h∗ω(A + X ) = ω(±A) = ±ω(A + X ),
wededucethath∗ω =±ω.Nextforany X ,Y ∈ ker ω,
dω(JX , Y) = g(X , Y) = g(h∗X , h∗Y) = dω(Jh∗X , h∗Y) = dω(h∗(h−1∗ Jh∗)X , h∗Y) = h∗dω((h−1∗ Jh∗)X , Y) =±dω((h−1∗ Jh∗)X , Y).
Bythenon-degeneracy oftheLeviformdω◦ J itfollowsthat
h∗J =±Jh∗ on ker ω. (2.4)
2.3. Joinof regularSasakimanifolds
WedescribeindetailanaturalprocedurewhichexplicitlyconstructsanewSasakimanifoldfromapair of given regular Sasaki manifolds. This corresponds to a variant of the join construction as is discussed in[10] for thecompact case. Inourcontext we apply the joininthe constructionof homogeneous Sasaki manifolds.
2.3.1. Sasaki immersions
LetX,Y beregularSasakimanifoldswithpseudo-Hermitianstructures{ω,J},{η,I},respectively.Also, letA,B denotetherespectiveReebvectorfieldsonX,Y .Animmersionofmanifolds
ι : Y → X
suchthat
i) the ReebvectorfieldA istangenttotheimageι(Y )⊆ X and
ii) thetangentbundleof ι(Y ) satisfiesJ T ι(Y )⊆ T ι(Y )
iscalled aSasaki immersion if
iii) {η,I}= ι∗{ω,J}
is satisfied.Thatis,for aSasaki immersion,{η,I} isobtainedby pullbackof {ω,J}.Letq : X → W and p: Y → V denotetherespectiveKählerquotients.ThentheSasakiimmersionι inducesauniqueKählerian immersion
j : V → W ,
suchthatj◦ p= q◦ ι.Notealsothatj determinestheSasakiimmersionι uniquely uptocompositionwith anelementoftheReebflowT .
2.3.2. The joinconstruction andSasaki immersions
Let
(Xi,{ωi, Ji}) , i = 1, 2 , beregularSasakimanifoldswithReebflows
Ti={φi,t}t∈R. Furthermore,let(Wi,Ωi) denotetheKählerquotientsofXi,and
qi: Xi→ Wi thecorrespondingBoothby-Wangfibrations.Nowconsider
¯
T = T1× T2={(φ1,s, φ2,t)}s,t∈R
anddefine Δ={(φ1,t,φ2,−t)}t∈Ras thediagonalinT .¯ Thenput
T = ¯TΔ .
Proposition2.4(Joinof Sasaki manifoldsX1 andX2).ThereexistsauniqueregularSasaki manifold X = X1∗ X2
withReeb flow T andKählerquotient
q : X1∗ X2 → W = (W1× W2, Ω1× Ω2) , whichadmitsSasaki immersions ιXi : Xi → X1∗ X2 suchthatthediagram
Xi X1∗ X2 Wi W1× W2 ιXi qi q pri (2.5) iscommutative(i= 1,2).
Proof. Observethat,viatheproductaction,T = T¯ 1×T2actsproperlyandfreelyonX1×X2withquotient
map
¯
q = q1× q2: X1× X2 → W = W1× W2.
Defineanotherquotientmap
p : X1× X2 → X := (X1× X2)/Δ , (2.6)
andlet
q : X→ W
betheinducedmapsuchthatq = q¯ ◦ p.
Letpri: X1× X2→ Xi,i= 1,2,denotetheprojectionmaps. Define
¯
ω = pr∗1ω1+ pr∗2ω2,
andconsider theKählerformΩ= Ω1× Ω2onW . Byconstruction,
¯
q∗(Ω) = d¯ω .
NextletA¯i denotethecanonicalliftsoftheReebfieldsAi toX1× X2,whereA¯iistangenttothefactor Xi,respectively.Theone-parametergroupsgeneratedbythesevectorfieldsarecontainedintheabelianLie groupT .¯ Inparticular,thesevectorfields areΔ-invariant.LetVΔdenote theone-dimensionaldistribution
onX1× X2,whichisspannedbythevectorfieldA¯1− ¯A2.ThenVΔisvertical(tangenttothefibers)with
respectto thequotientmapp in(2.6) inducedbytheactionofΔ.Therefore,bothvectorfields A¯iproject tothesamevectorfieldA onX.
Note that ω is¯ a T -invariant¯ one-form which vanishes on VΔ. Therefore, there exists on X a unique
inducedoneform
ω = ω1∗ ω2 satisfyingp∗ω = ¯ω.
Inparticular,ω satisfiesq∗Ω= dω,where Ω= Ω1× Ω2.Itfollows thatω isacontactformwithReebfield A.TheReebflowofω istheone-parametergroup
T = ¯T /Δ .
Summarizingtheconstruction,wenote thatω isaconnectionformfortheT -principalbundle q : X → W
andithascurvatureformΩ.
Let Ji denote the complex structures on ker ωi (canonically extended to tensors on Xi by declaring
Ji(Ai)= 0).Observethatthekernel ofω coincideswiththeprojectionof ker pr∗1ω1∩ ker pr∗2ω2
to(thetangentbundleof)X.Therefore,J = J¯ 1× J2goesdowntoanalmostcomplexstructureJ onker ω
suchthat
q : (X,{ker ω, J}) → (W, J)
isaholomorphicCR-map.Since(W,{Ω,J}) isKählerandω aconnectionformwithcurvatureΩ,thealmost CR-structure{ker ω,J} isintegrable,see[17,Theorem 2].Since
dω◦ J = (q∗Ω)◦ J
ispositive,{ω,J} definesapseudo-Hermitian structureonX.BytheconstructionT actsby holomorphic transformationsonX.Thisshowsthat(X,{ω,J}) isaregularSasakimanifoldwithKählerquotient(W,Ω). Choose a base point (xo,yo) ∈ X1 × X2 and define immersions ιi : Xi → X, ι1(x) = q(x,yo) and ι2(y) = q(xo,y). (Note that all such pairs of maps are equivalent by an element of T .) By the above construction, ιi are Sasaki immersions, and, in fact, they determine the Sasaki structure {ω,J} on the manifoldX1∗ X2 uniquely,together withtheconditionthatA istheReebfield.
ThejoinofSasaki manifoldsenjoysthefollowing functorialproperty:
Proposition2.5. Foranypairof Sasaki immersionsτi: Yi→ Xi with inducedKähler immersionsji : Vi→
Wi,i= 1,2, thereexistsauniqueSasaki immersion
τ = τ1∗ τ2: Y1∗ Y2 → X1∗ X2 suchthattheassociated diagram
Y1∗ Y2 X1∗ X2 V1× V2 W τ p q j1×j2 (2.7)
iscommutativeand ιXi◦ τi andτ◦ ιYi coincideuptoan element ofT .
Proof. Sinceτi areSasakiimmersions,theproduct map ¯
τ = τ1× τ2: Y1× Y2→ X1× X2
inducesamap
τ1∗ τ2:= Y1∗ Y2→ X1∗ X2
withtherequiredproperties. This gives:
Corollary2.6.The join ofX1 andX2 definesanatural homomorphism
Psh (X1)× Psh (X2) → Psh (X1∗ X2) , (φ1, φ2) → φ1∗ φ2 withkernel thediagonal groupΔ={(φ1,t,φ2,−t)}t∈R.
Proof. Indeed,bytheconstructioninProposition2.5,φ1∗ φ2∈ Psh (X) andtheabovemapisa
homomor-phismwithkernelΔ. Wecall thegroup
Psh (X1)∗ Psh (X2) = (Psh (X1)× Psh (X2))
Δ
thejoin ofthegroupsPsh (Xi).Bytheabove,thejoinofPsh (Xi) identifieswithasubgroupofPsh (X1∗X2).
Corollary2.7.LetX1andX2behomogeneousSasakimanifolds.ThenthejoinofgroupsPsh (X1)∗Psh (X2) isacting transitively by pseudo-Hermitian transformations on the Sasaki manifoldX1∗ X2.In particular, X1∗ X2 isa homogeneousSasakimanifold.
Proof. TheKählerquotientW1×W2ofX1∗X2isahomogeneousKählermanifoldforthegroupG= G1×G2,
whereGi denotes theBoothby-Wang imageof Psh (Xi) inIsomh(Wi).Since G isalso theBoothby-Wang imageofPsh (X1)∗Psh (X2),andthelatteralsocontainstheReeb-flowT ,itfollowsthatPsh (X1)∗Psh (X2)
actstransitivelyonX1∗ X2.
3. Pseudo-HermitiangroupPsh (X) ofaregularSasakimanifoldwithvanishing Kählerclass
Suppose that(X,{ω,J}) is aregular Sasaki manifold with Reeb flow T isomorphic to the real line R. ThentheBoothby-WangfibrationProposition2.2givesaprincipalbundle
R −−−−→ X −−−−→ Wq
overtheKähler quotientW = (W,Ω,J).Here thegroupR={ϕt}t∈R oftheprincipalbundleisgenerated bytheReebfieldandtheKählerformonthebase issatisfyingtheequation
q∗Ω = dω . (3.1)
Chooseasmoothsection s: W → X ofq such thatthebundle X isequivalent tothetrivialbundle by abundlemap
f :R × W −→X
whichisdefinedby
f (t, w) = ϕts(w) . Wethushavethefollowing commutativediagram:
R × W −−−−→f X
pr q
W
, q◦ s = idW . (3.2)
Declareaone-formθ onW byputting
θ = s∗ω . (3.3)
Notethenthatdθ = Ω from(3.1).Inparticular,theKählerformΩ on W is exact.
Nextextendθ toatranslationinvariantone-formonR× W bydeclaring
ω0= dt + pr∗θ, so that dω0= pr∗Ω holds. (3.4)
Notingf (0,w)= s(w)= s◦ pr(0,w),wehave
pr∗θ|{0}×W = ((s◦ pr)∗ω)|{0}×W = f|∗{0}×Wω.
Sinceboth formsf∗ω andω0 aretranslationinvariant,weconcludethat
f∗ω = ω0. (3.5)
Thenanalmostcomplexstructure J on˜ ker ω0 isdefinedby
pr∗◦ ˜J = J◦ pr∗. (3.6)
Byconstruction,theisomorphismf∗: ker ω0→ ker ω isholomorphic,thatis,
f∗◦ ˜J = J◦ f∗.
Inparticular,J is˜ acomplexstructureonker ω0.Summarizingtheaboveweobtain:
Proposition3.1.IdentifyingtheregularSasakimanifoldX withR×W viaf ,thepseudo-Hermitianstructure {ω,J,A} corresponds to{ω0,J,˜ ∂t∂} onthetrivialbundleR× W ,where ω0 isdefined asin(3.4).
Existenceof acompatibleregularSasaki manifold. Conversely,anyexact Kählerform Ω = dθ
onacomplexmanifoldW arisesasthecurvatureformofaconnectionform ω onthetrivialprincipalbundle
X =R × W .
In fact, such ω with Reeb field A = ∂t∂ is given by (3.4). As aconsequence (employing [17, Theorem 2] to show the integrability of the almost CR-structure {ker ω,J}), there exists on X a pseudo-Hermitian structure
{ω, J, A} , (3.7)
whichhas the Kähler manifold (W,Ω) asits Kähler quotient. We call such apseudo-Hermitian structure
compatiblewiththeKähler manifold(W,Ω).
We remark now that, under a mild assumption on the Kähler manifold W , any compatible pseudo-HermitianstructureonX isessentiallydetermined uniquelybytheKähler structureonW .
Proposition 3.2. Suppose H1(W,R) = {0}. Then any two pseudo-Hermitian structures {ω,J,A} and {ω,J,A} on X, which are compatible with the Kähler manifold (W,Ω), are related by a gauge
trans-formationfortheprincipalbundleq : X → W .
Proof. Bythecompatibilityassumption,wehaveω− ω = q∗η,forsomeclosedone-formη∈ Ω1(W ).Since
H1(W,R)={0},there exists afunctionλ: W → R suchη = dλ. Intheviewof Proposition3.1, wemay
assumethat X =R× W andω = dt+ q∗θ,where dθ = Ω. We define agauge transformation G for the bundleq,byputting
G(t, w) = (t + λ(w), w) .
Wethen calculateG∗ω = G∗dt+ q∗θ = dt+ dq∗λ+ q∗θ = ω + q∗η = ω.
Remark 3.3.For an analogous existence result for Sasaki manifolds in the more elaborate case of circle bundlesoverHodgemanifolds,see[8,Theorem3],respectively[21].
3.1. Lifting ofisometriesfrom theKählerquotient
We now proveastructure result forthegroupof holomorphicisometries Psh (X) of X ifthe Boothby-Wangfibrationhas contractible fiberR.Thatis, letX be aregularSasaki manifold with Boothby-Wang fibration
R → X → W . (3.8)
Asbeforelet
Isomh(W ) = Isom (W, Ω, J)
denotethegroupofholomorphicisometriesoftheKählerquotientW forX.
Proposition 3.4. Assume that the first cohomology of the Kähler quotient W , arising in (3.8), satisfies H1(W )={0}. ThentheBoothby-Wanghomomorphism (2.2) definesanaturalexact sequence
1 −−−−→ R −−−−→ Psh (X) −−−−→ Isomφ h(W ) −−−−→ 1 . (3.9)
In particular,Psh (X) acts transitivelyonX if andonly ifIsomh(W ) acts transitivelyonW .
Proof. IntheviewofProposition2.2itissufficienttoshowthatφ issurjective.Indeed,sinceH1(W )={0},
Lemma 3.5belowshowsthatfor anyh∈ Isomh(W ),there exists anisometry ˜h∈ Psh (X),which isalift ofh,thatis, φ(˜h)= h.
Proposition 3.4 is implied by the following basic lifting result for holomorphic and anti-holomorphic isometriesoftheKählerquotient W :
Lemma 3.5. Assumethat H1(W )={0}, andlet h∈ Isom (W ) satisfy h∗Ω= μΩ, where μ∈ {±1}. Then
there existsan isometryh˜∈ Psh±(X) suchthat h induces˜ h on W andsatisfies ˜h∗ω = μω. Ifμ= 1 then ˜
h∈ Psh (X).
Proof. We may assume X = R × W . Define ˜h(t,w) = (t,h(w)) to be the canonical lift of h. Then
ω = μ· (˜h)∗ω defines another pseudo-Hermitian structure on X which is compatible with (W,Ω). By Proposition3.2,thereexists agaugetransformationG: X→ X withG∗ω = ω.Therefore,
˜
h = G◦ ˜h
satisfies˜h∗ω = μ·ω,anditisanisometricliftofh forthemetricg = ω·ω +dω ◦J.Italsofollows˜h∗A= μA. Thus,˜h∈ Psh±(X).
4. HomogeneousSasakimanifolds
SupposethattheLiegroupG actstransitivelybypseudo-HermitianisometriesontheSasakimanifold X. Then
X = G/H
iscalled ahomogeneousSasaki manifold.SinceX isalsoacompleteRiemannianmanifoldwith respectto theSasaki metricg,the ReebfieldA for X,whichis aKillingfieldforthe metricg,is acompletevector field.Let
T ={ϕt}t∈R
denotethe1-parameter grouponG/H generatedbytheReebfield.
4.1. NaturalfiberingoverhomogeneousKähler manifold
Since T commuteswithG,there existsaone-parameter subgroup
A = {at}t∈R ≤ NG(H) (4.1)
suchthat
ϕt(xH) = xa−1t H , (4.2)
whereNG(H) denotesthenormalizerofH inG.
Proposition4.1. T isaclosed subgroup inPsh (G/H). Inparticular, T isisomorphic toS1 or R,anditis actingproperly onG/H.
Proof. TheReebfieldA isuniquelydeterminedbytheequations:
ω(A) = 1, ιAdω = 0.
LetD ={ϕt}t∈R≤ Psh (G/H) betheclosure.AsLgϕt= ϕtLg (for all g∈ G) from(4.2),everyelementof
D commuteswith G.Thusevery vectorfieldB inducedfromone-parameter groupsinD is left-invariant. Inparticular,ω(B) is constant.BytheCartanformula,it followsιBdω = 0. Ifω(B)= 0,byuniquenessof theReebfield,B = A upto aconstantmultipleonG/H.Whenω(B)= 0,thenon-degeneracy oftheLevi formdω◦ J onker ω implies B = 0 onG/H. ThisshowsD ={ϕt}t∈R.
Lemma4.2. T actsfreely onG/H.
Proof. Ifϕt0(x0H)= x0a−1t0 H = x0H,forsomex0∈ G,thenat0 ∈ H andsoϕt0(xH)= xH (for all x∈ G).
SinceT actseffectively,ϕt0= 1.
Inparticular,anyhomogeneousSasakimanifoldX = G/H isaregularSasakimanifold(cf.[8]).Moreover, byProposition2.2theKählerquotient
W = (G/H)T
isahomogeneousKählermanifoldforG.Thatis,G isactingtransitivelybyholomorphicisometriesonW .
Wethushave:
Theorem 4.3 (Boothby-Wang fibration [8]).Every homogeneous Sasaki manifold X = G/H arises as a principalT -bundleoverahomogeneousKähler manifoldW which takestheform:
T −−−−→ G/H −−−−→ W = G/HA .q (4.3)
Remark4.4.IfG/H iscontractible, soisG/HA,and inthiscaseT ∼=R.
The following existence and uniqueness result for contractible homogeneous Sasaki manifolds is now a directconsequenceofSection3:
Corollary4.5(ContractiblehomogeneousSasakimanifolds).Let(W,Ω,J) beahomogeneousKählermanifold which iscontractible. Then there existsa contractiblehomogeneous Sasaki manifold(X,{ω,J}) which has Kählerquotient(W,Ω,J).Moreover,withtheseproperties,theBoothby-Wangfibration(4.3) forX hasfiber
R,and X isuniquelydefined uptoapseudo-Hermitian isometry.
Proof. Indeed,wemaychooseonthetrivialprincipalbundleX =R× W ,thepseudo-Hermitianstructure (3.7), which has Reeb field A = ∂t∂ and Kähler quotient (W,Ω,J). By Proposition 3.4, (X,{ω,J,A}) is
ahomogeneous Sasaki manifold. Let (X,{ω,J,A}) beanother contractible Sasaki manifold which has (W,Ω,J) asaKählerquotient.ThentheBoothby-WangfibrationforXhasfiberR,and,byProposition3.1, thereexistsapseudo-HermitianisometryfromX to(X,{ω,J,A}).ByProposition3.2,thelatteradmits apseudo-Hermitian isometry to(X,{ω,J,A}) whichis given byagaugetransformationof thebundle X.
Thisimpliestheclaimeduniqueness.
4.2. Pseudo-Hermitianpresentations of W
LetX beahomogeneousSasakimanifoldwithgroupG andW itsKählerquotient.Wedescribenowthe typesofhomogeneouspresentations
W = G/HA
whichcanarise intheassociatedBoothby-Wang fibration(4.3).Forthiswe assumethat
G ≤ Psh (X)
is aclosed subgroup. Inparticular,G is acting faithfullyon X.With this assumption the stabilizer H is
alwayscompact,sinceG isaclosed groupofisometriesforX.
Lemma 4.6.Let Δ denote the kernel of the induced G-action on the Kähler quotient W of X. Then the followinghold:
(1) HA= H A decomposesasasemi-direct product.
(2) Δ≤ HA, and,L = H¯ AΔ iscompact.
(3) Δ= T∩ G,in particular,Δ iscentralinG.
(4) If A is non-compact then theprojection homomorphismπA : HA→ A maps Δ injectively toa closed subgroupof A.
(5) IfA is normalin G thenA is centralinG.
Proof. SinceT actsfreelyonG/H,weinferfrom(4.2) that A∩ H = {1}.Thisimpliesthat
HA = H A
isasemi-directproduct,proving(1).Let
πA: HA → A
denote the projection homomorphism. Since H is compact, the homomorphism πA is proper. Therefore,
the image G of¯ G in Isom (GHA) is closed and acts properly on W = G/HA = ¯G/ ¯L. We deduce that ¯
L = HAΔ isacompactsubgroupofG = G/Δ.¯ Thus,(2)holds.
Since thehomomorphismφ in(2.2) whichmapsG toG has¯ kernelT ,
Δ = G∩ T ,
where theintersectionis takeninPsh (X). Recall thatT is central inPsh (X).Therefore, Δ iscentralin
G.Hence,(3).
Next, consider C = ker πA∩ Δ= H∩ Δ. AssumingthatA is avectorgroup,C is theunique maximal
compact subgroup of Δ.Since Δ is normal in G, so is C. Since C is also a subgroup of H and G/H is
effective, we deduce thatC = {1}.This shows thatΔ is isomorphic to the closed subgroup πA(Δ) ≤ A,
proving(4).
Finally,assumethatA isnormalinG.Thentheleft-multiplicationorbitsofA onG/H coincidewiththe orbitsofT .Thatis,forallg∈ G:
T · gH = g · AH = A · gH .
In particular,the left-action of A on G/H (which is by pseudo-Hermitian isometries) induces the trivial actionontheKähler-quotient W bythe fibrationsequence(4.3).Thatis, A≤ Δ and by(3), A≤ T . This impliesthatA= T iscentralinG.
Twoprincipalcasesarearising,accordingtowhetherΔ isacontinuousgrouporΔ isadiscretesubgroup ofG.RecallfirstthateitherA= S1 orA=R.Thenwehave:
CaseI (Δ = A,T is containedinG). We suppose herethatA can be chosento be anormal subgroup in
G.By(5) ofLemma4.6, itfollowsthattheisometriesinducedbytheleft-actionof A arecontainedinthe kernelofthehomomorphismφ: Psh (X)→ Isomh(W ),whichisjustT .SinceA isanon-trivialconnected (one-dimensional)group,thisimplies
T =A = Δ
as subgroupsof Psh (X). Then thefibration(4.3) turns into aprincipalbundle ofhomogeneous spacesof theform
A −−−−→ G/H −−−−→ W = (G/A) /H = ¯q G/ ¯H , (I)
whereH = H and¯ thegroupG is¯ describedbyanexactsequence ofgroups
1 −−−−→ A = R −−−−→ G −−−−→ ¯φ G −−−−→ 1 . (I)
CaseII(A = R, Δ = Z). WeareassumingthatA=∼R (forexample,ifG/H iscontractible).ByLemma4.6 (4), the central subgroup Δ of G is either infinite cyclic (and discrete) or Δ is a closed one-parameter subgroupinHA whichisprojecting surjectivelyontoA.SinceΔ iscontainedinT ,andT isonedimensional, wededuceΔ= T ,inthelattercase.This situationwasalreadydescribedinCaseIabove.
So for case II, Δ = T ∩ G is infinite cyclic and central in G. Moreover, Δ ≤ HA and by Lemma 4.6 (4)themapπA is projectingΔ injectivelyontoadiscretelatticeZ inA.Denote withA the¯ imageofA in
¯
G = G/Δ.ThentheBoothby-Wangfibration(4.3) can bewrittenintheform
A −−−−→ G/H −−−−→ W = ¯q GH ¯¯A , (II)
wherethegroupG is¯ described bytheexactsequence
1 −−−−→ Δ = Z −−−−→ G −−−−→ ¯φ G −−−−→ 1 . (II)
Recall also that L = ¯¯ H ¯A is a compact subgroup of G,¯ and H is¯ a compact normal subgroup in H ¯¯A.
Therefore,thesimplyconnectedone-parametergroupA maybechoseninsuchawaythatitsquotientA is¯ acompactcirclegroup,and theintersectionH¯∩ ¯A is finite.
5. HomogeneousKählermanifoldsofunimodulargroups
LetW be ahomogeneousKähler manifold. Thefundamentalconjecture forhomogeneous Kähler man-ifolds(as proved by Dorfmeisterand Nakajima [12])asserts thatW is aholomorphic fiberbundle over a homogeneousboundeddomainD withfibertheproductofaflatspaceCk withacompactsimplyconnected homogeneousKählermanifold.
Recall thataLie groupG iscalled unimodular ifitsHaar measure isbiinvariant. Letg denote theLie algebra of G. If G is connected, then G is unimodularif and only if the trace function over the adjoint representationofg iszero.
Proposition5.1. LetW be acontractiblehomogeneousKählermanifoldthatadmits aconnectedunimodular subgroup
G≤ Isomh(W )
whichactstransitively onW .Thenthere existsasymmetricboundeddomain D such W =Ck× D
isaKähler directproduct.
AmoregeneralresultforarbitraryhomogeneousKählermanifoldsofunimodularLiegroupsisstatedin [1,Proposition4.2] andProposition5.1canbederivedasaspecialcase.Forclarityandcompletenessofthe expositionwegiveadetailedandselfcontainedproof.TheconditionthatahomogeneousKähler manifold iscontractibleclarifiestheexistenceofsymmetricdomainintheproductdecompositionandalsosimplifies theproof.Thisresultwill beusedto determineisometrygroupsinthenextsection.
Proof of Proposition5.1. For the proof we require some constructions which are developed in the proof of the fundamental conjecture as it is given in[12]. The first main step inthe proof of the fundamental conjectureistomodifyG inordertoobtainasuitableconnectedtransitiveLiegroupG withˆ particularnice properties[12,Theorem2.1].ByamodificationprocedureonthelevelofLiealgebras(asisdescribedin[12, §2.4]),we obtainfromtheKähler Liealgebra g ofG aquasi-normal KählerLie algebraˆg.Moreover, itis shownthatthereexistsaconnectedsubgroupGˆ≤ Isomh(W ),whichhasLiealgebraˆg andactstransitively onW . As can be verifieddirectlyfrom[12,§2.4],the modifiedLie algebraˆg preserves unimodularityof g andalsosatisfies dim ˆg ≤ dim g.
Therefore, from thebeginning, we may assume thatthe connected unimodulartransitive Lie group G
of holomorphic isometries in question has a quasi-normal Lie algebra g. We can also replace G with its universalcoveringgroup,andweremarkthatK is connected (W = G/K issimplyconnected,sinceweare assumingherethatW iscontractible).Withtheseadditionalpropertiesinplace,accordingto[12,Theorem 2.5] combinedwith [12,§7],thefollowinghold:
(1) There exists a closed connected normal abelian subgroup A of G, such that G = AH is an almost semi-directproduct.
(2) Thereexistsareductive subgroupU ≤ H,with K≤ U,suchthat
D = H/U
isaboundedhomogeneousdomainand
U/K
iscompactwithfinite fundamentalgroup.
(3) PutL= AU .ThenL isaclosedsubgroupofG andthemap
W = G/K → G/L = H/U = D
isaholomorphicfiberbundle withfiberL/K = AU/K.
We prove now that, if G is unimodular then H is a unimodular Lie group: For this recall from [12, Theorem2.5] thatA istangenttoaKähler ideala oftheKähleralgebrawhichbelongstoW . (Recallthat theKähleralgebraforG/K isg togetherwith analternatingtwo-formρ whichisrepresentingtheKähler formonW .)SinceK intersectsA onlytrivially,theKählerideala isnon-degenerate,thatis,therestriction
ρa oftheKählerformρ ofg toa isnon-degenerate.Sincea isabelianandρ isaclosedformong,itfollows that ρa is invariant by the restriction of the adjoint representation of h (respectively H). In particular,
thisrestrictedrepresentationofH ona isbyunimodularmaps. SinceG isunimodular,itfollows fromthe semi-directproduct decompositionG= AH that H isunimodular.
LetK1denotethemaximalcompactnormalsubgroupofH.ThenthegroupH= H/K1 isunimodular.
Moreover H acts faithfully and transitively on D = H/K = H/K, K = K/K1. Hence, the bounded
domainD hasatransitivefaithfulunimodulargroupH ofisometries.ByresultsofHano[13,TheoremIII, IV],H must be semisimpleand D is asymmetric bounded domain. Wealso concludethatthere exists a semisimplesubgroupS≤ G,whichisofnon-compacttype,suchthatH = K1S isanalmostdirectproduct
andthehomomorphismS→ H isacoveringwithfinite kernel.
Contractibility ofW furtherimpliesU = K.Therefore,inthiscase,theholomorphicbundlein(3)isof theform
G/K → D = H/K ,
withfiberA=Ck,andD isasymmetricboundeddomain.
Finally the direct product decomposition follows: Note also that K1 acts faithfully on Ck by Kähler
isometries, and that S acts trivially on A, since it is of non-compact type. It follows thatS is a normal subgroupofG.Thereforeits tangentalgebramustbe orthogonalto A withrespect toρ.Since theKähler algebrag belongingto G isdescribing W ,we concludethatthere isanorthogonal productdecomposition
W =Ck× D.
Wealsoobtain:
Corollary5.2. SupposethatW isacontractiblehomogeneousKählermanifold,andthatthereexistsadiscrete uniformsubgroupinIsomh(W ).ThenW is Kählerisometric toCk× D, whereD isasymmetricbounded domain.
Thefollowing isobtainedintheproofofProposition5.1:
Corollary 5.3. Assume that W is a homogeneous Kähler manifold which admits a transitive unimodular groupG.Thenthereexistsasymmetricboundeddomain D suchthatW is aholomorphicfiberbundleover D with fibertheproductof aflat spaceCk withacompactsimplyconnectedhomogeneousKählermanifold. Moreover, Isomh(W )0 contains a covering group of the identity component of the holomorphic isometry groupof D.
Proof. In fact, in the proof of Proposition 5.1 it is established that D is symmetric with a semisimple transitivegroupS contained inthe quasinormalmodification G of¯ G.It is alsoclearthatS is normalin
¯
G,anditisthemaximalsemisimplesubgroupofnon-compacttypeinG (in¯ fact,inIsomh(W )0),andS is
coveringIsomh(D)0.
Werecallthatany symmetricboundeddomainD admitsaninvolutiveanti-holomorphicisometry: Proposition 5.4 (Isometry group of symmetric bounded domain).Let D be a symmetric bounded domain with Kählerstructure(Ω,J).If D is irreducible then
Isom (D) = Isomh±(D) .
Moreover, foranyD there existsanelement τ¯∈ Isom (D) such that
¯
τ2= 1, ¯τ∗Ω =−Ω, ¯τ∗J =−J ¯τ∗.
Forthe factthateveryisometry of an irreduciblebounded symmetric domain is eitherholomorphicor anti-holomorphic, see e.g.[18, Ch. VIII, Ex. B4]. For the existence of the anti-holomorphic involutionτ ,¯ recallfirst thatthe metriconanysymmetric boundeddomain D isanalytic (see[18]).Then thefollowing holds:
Proposition 5.5. Let W be a simply connected Kähler manifold with analytic Kähler metric. Then there existsan anti-holomorphicinvolutiveisometry ¯τ ofW .
Proof. Since W is a complex manifold and the Kähler metric is Hermitian with respect to the complex structure,thereexistlocal complex coordinatesforW suchthatthemetriccan bewrittenas
g0= 2
α,β
gα ¯βdzαd¯zβ,
where gα ¯β is a Hermitian matrix, so that gα ¯β = gβ ¯α. In particular, the Kähler form Ω0 is obtained as
Ω0=−2i
α,β
gα ¯βdzα∧ d¯zβ.
Letτ0:Cn→Cn bethecomplexconjugationmap,thatis,
τ0(z1, z2, . . . , zn) = (¯z1, ¯z2, . . . , ¯zn).
Thenτ0 satisfiesτ0∗Ω0=−Ω0, τ0∗JC=−JCτ0∗.Inparticular,τ0 definesalocalanti-holomorphicisometry
ofW .
Since W is simply connected, we may use analytic continuation to extend τ0 to an analytic map τ :¯ W → W . By the analyticityassumptions, τ is¯ an anti-holomorphic map and it is preserving the Kähler metric.Alsoitfollows¯τ2= id
W bythelocalrigidityofanalyticmaps.Therefore,¯τ isaninvolutiveisometry of W .
Remark5.6.NotethattheholomorphicisometrygroupIsomh(D) hasfinitelymanyconnectedcomponents. Interestingly,evenifD isirreducibleIsomh(D) isnotnecessarilyconnected[18,Ch. X,Ex.8].
6. LocallyhomogeneousasphericalSasakimanifolds
Inthissection X denotesaregularcontractibleSasakimanifold.
6.1. HomogeneousSasaki manifoldsforunimodulargroups
Since X is regular with Reeb flow isomorphic to the real line, Proposition 3.4 implies that the Reeb fibering
R → X−→ Wq
givesrisetoanexactsequenceofgroups
1→ R → Psh (X)−→ Isomφ h(W )→ 1 , (6.1) whereW istheKählerquotientof X.
Proposition6.1. LetX be a homogeneousSasaki manifold suchthat itsKähler quotient W iscontractible. Supposefurtherthat X admitsaconnectedtransitiveunimodularsubgroup
G ≤ Psh (X) . Thenthefollowinghold:
(1) W =Ck× D istheKähler productof aflat spacewithasymmetricbounded domainD.
(2) If theReebflow of X isisomorphic toR,thenthepullback of
Ck U(k) ≤ Isomh(W )
along theexact sequence(6.1) isanormal subgroup
N U(k) ≤ Psh (X) , where N isa 2k + 1-dimensionalHeisenbergLiegroup.
Proof. As the Reeb flow T is central in Psh (X)0, the associated Boothby-Wang homomorphism φ as in
(6.1) mapstheunimodulargroupG to
¯
G = φ(G) ≤ Isomh(W ) .
Since also G is¯ unimodular and transitive onthe contractible Kähler manifold W , Proposition5.1 states thatW =Ck× D,whereD isasymmetricboundeddomain.Thisproves(1). Italsofollowsthat
Isomh(W ) =Ck U(k)× Isomh(D) .
WemaythuspullbackthefactorCk U(k) byφ intheexactsequence(6.1).Aspullbackweobtainthe subgroupN U(k) ≤ Psh (X),where N isthepreimageofthetranslationgroupCk.
Assuming T = R,we note that N is a centralextension of the Reeb flow R by the abelian Lie group Ck. We prove now thatN is a 2k + 1-dimensional Heisenberg Lie group by showing thatits Lie algebra n has one-dimensional center:Since N actsfaithfully as atransformation grouponX, wemay identifyn withasubalgebra ofpseudo-Hermitian Killingvectorfields onX.Thissubalgebra containstheReebfield
A (tangentto thecentral one-parametergroupT = R)inits center.Now, sinceN is thepullbackof Ck, givenany twovectorfieldsX ,Y ∈ n,wehave
[X , Y] = ω([X , Y])A . UsingLemma 6.2below,weobserve
q∗Ω (X , Y) = dω(X , Y) = ω([X , Y]) .
Since(Ck,Ω) isKähler,itfollowsthatdω definesanon-degeneratetwo-formonn/A.Thisshowsthatthe Liealgebran hasone-dimensional centerA.Thereforethe LiegroupN hasone-dimensional center.SoN isaHeisenberggroupofdimension2k + 1.
A vector field on X with flow in Psh (X) will be called a pseudo-Hermitian vector field. The set of pseudo-Hermitian vectorfields forms asubalgebra ofthe LiealgebraofKilling vectorfields fortheSasaki metric g.
Lemma6.2. LetX ,Y be anytwopseudo-Hermitian Killingvector fieldson theSasakimanifoldX. Then q∗Ω (X , Y) = dω(X , Y) = ω([X , Y]) .
Proof. Sincetheflow ofX preservesthecontactformω,wehave
LXω = 0 . (Here,LX denotestheLiederivativewithrespectto X .)Thatis,
LXω(Z) − ω([X , Z]) = 0, forallvectorfieldsZ onX.Wecompute
0 = LXω(Y) − ω([X , Y]) − LYω(X ) + ω([Y, X ]) = LXω(Y) − LYω(X ) − ω([X , Y]) + ω([Y, X ]) = dω(X , Y) − ω([X , Y]) .
LetX beacontractiblehomogeneousSasakimanifoldwithKählerquotientW =Ck× D,whereD isa symmetricboundeddomain.Then
Isomh(W ) =Ck U(k)× Isomh(D) . (6.2)
Note further that Isomh(D)0 = S
0 is the identity component of the holomorphic isometry group of a
Hermitiansymmetricspace
D = S0/H0
of non-compact type. Inparticular,S0 is semisimple of non-compact type[18, Ch. VIII,§7] and without
center.Therefore(6.2) alsogives:
Proposition6.3. Psh (X) hasfinitely manyconnectedcomponentsand
Psh (X)/ Psh (X)0= Isomh(D)/ Isomh(D)0.
Weadd:
Proposition6.4(Sasakiautomorphismgroup).Thereexistsasemisimple LiegroupS of non-compact type, whosecenterΛ isinfinitecyclic,anda2k +1 dimensionalHeisenberggroupsN ,suchthatthereisanalmost directproductdecomposition
Psh (X)0 = (N U(k)) · S . Moreover,theReeb flowT of X isthecenterof N and
T∩ S = (N U(k)) ∩ S = Λ (∼=Z).
Proof. ForthehomogeneousSasakimanifoldX,theexactsequenceofgroups(6.1) associatedto theReeb fiberingforX inducesacentralextension
1→ T → (N U(k)) · S → (Cφ k U(k)) × S0→ 1 , (6.3)
wheretheReebflowT =R mapsto thecenterofN .Here
S≤ Psh (X)0
isasemisimplenormalsubgroupofnon-compacttype,whichiscoveringS0underφ.Inparticular,sinceS
isanormalsubgroup ofPsh (X)0,it commutes withN U(k).(Notealso thatU(k) actsfaithfullyonN
andmapstoamaximalcompactsubgroupofAut(N ).) Thekernel Λ ofthecoveringS→ S0 is
ker φ∩ S = R ∩ S = (N U(k)) ∩ S .
Moreover, Λ isthe center of S, since S0 has trivialcenter. We claim that Λ is an infinite cyclic discrete
subgroupand,inparticular,itisauniformsubgroupinT .Indeed,inthelightofCorollary4.5,thereexists aunique contractible homogeneous Sasaki manifold X1 with Kähler quotient Ck, and similarly aunique
homogeneousSasakimanifold X2 with Kähler quotientD.Let Ti ≤ Psh (Xi) denotetheReebflow ofXi.
Then(seeSection2.3,Corollary2.7)thejoinX1∗X2isahomogeneousSasakimanifoldwithKählerquotient
Ck× D. Accordingtotheabove,Psh (X
1)=N U(k),and byProposition6.9belowPsh (X2)0= T2· S,
whereS isaclosedsemisimple LiesubgroupcoveringS0 with infinitecyclickernelΛ= Z(S),T2∩ S = Λ.
ItfollowsthatPsh (X)0= Psh (X
1)∗ Psh (X2)0hastheclaimedproperties. 6.2. ApplicationtolocallyhomogeneousSasaki manifolds
Weconsider acompactasphericalSasaki manifoldoftheform
M = Γ\X ,
whereX is acontractibleSasaki manifold andΓ isatorsion freediscrete subgroupcontained inPsh (X). IfX isahomogeneousSasakimanifoldthenM iscalled alocallyhomogenousSasaki manifold.
Theorem6.5. Supposethat X isacontractible homogeneousSasaki manifoldandthat X admits adiscrete subgroup ofisometrieswith
Γ\X
compact. Then:
(1) TheKähler quotientofX isaKähler product
W =Ck× D ofaflat spaceCk with asymmetricboundeddomainD.
(2) LetT denotetheReebflow ofX. ThenΓ∩ T isadiscreteuniformsubgroupof T (inparticular,Γ∩ T
isisomorphic toZ).
(3) Let φ: Psh (X)→ Isomh(W ) betheBoothby-Wang homomorphismin (6.1).Thenthesubgroup φ(Γ) ≤ Isomh(W )
isdiscreteanduniform.
Corollary 6.6. Let M = X/Γ be a compact locally homogeneous Sasaki manifold. Then M is a Sasaki manifold with compact Reeb flow T = S1. Moreover, a finite covering space of M is a regular Sasaki manifold.
Remark 6.7. Certain linear flows on the sphere give rise to irregular compact Sasaki manifolds, cf. [11, Chapters2,7].
ForthepreparationoftheproofofTheorem6.5weshall recallsomestandardfactsabout:
Levidecompositionanduniformlattices. Ingeneralaconnected LiegroupG admitsaLevidecomposition G = R · S ,
where R is the solvable radical of G and S is asemisimple subgroup. Let K denotethemaximal compact and connected normal subgroup of S,then put S0 =G/(RK). Note thatS0 is semisimpleof non-compact
type.Wewillneedthefollowingfact(see[28,Chapter 4,Theorem 1.7],forexample):
Proposition6.8. LetΓ beauniformlatticeinG.Thentheintersection(RK)∩ Γ isauniformlattice inRK. In particular,intheassociatedexact sequence
1 −−−−→ RK −−−−→ G −−−−→ Sν 0 −−−−→ 1, (6.4) theimage ν(Γ) isa uniformlatticein thesemisimpleLie groupS0.
Remarkinaddition thefollowing:Asthesubgroup ν(Γ)≤ S0 isdiscrete anduniform,and sinceS0 has
nocompactnormalconnectedsubgroup,theimageofν(Γ) is aZariskidensesubgroupintheadjointform ofS0(byBorel’sdensitytheorem,cf.[24]).ConsideranyconnectedclosedsubgroupG ofS0,whichcontains ν(Γ).ThenG isuniformandZariski-dense.ThisimpliesthatG = S0.
Nowweareready forthe
