On a Gromoll-Meyer type theorem in globally hyperbolic stationary spacetimes

0.5 MB·English

by  L. Biliotti

ON A GROMOLL-MEYER TYPE THEOREM
IN GLOBALLY HYPERBOLIC STATIONARY SPACETIMES
LEONARDOBILIOTTI,FRANCESCOMERCURI,ANDPAOLOPICCIONE
7
0 ABSTRACT. Following the lines of the celebrated Riemannian result of Gromoll and
Meyer[21],weuseinfinitedimensionalequivariantMorsetheorytoestablishtheexistence
0
ofinfinitelymanygeometricallydistinctclosedgeodesicsinaclassofgloballyhyperbolic
2
stationaryLorentzianmanifolds.

n
a
J

3
2 CONTENTS

] 1. Introduction 1
G 2. Preliminariesaboutbilinearformsonnormedvectorspaces 6
D 3. Ontheindexofessentiallypositivebilinearforms 8
. 4. OntheMaslovindexanditerationformulas 12
h
4.1. MaslovandConley–Zehnderindex 13
t
a 4.2. PeriodicsolutionsofHamiltoniansystems 14
m
4.3. AniterationformulafortheMaslovindex 15
[ 4.4. MaslovindexofageodesicandofthecorrespondingHamiltoniansolution 16

5. Thevariationalsetup 17
1
v 5.1. Theconstrainedvariationalproblem 18
4 5.2. ThePalais–Smalecondition 19
5 6. TheMorseindextheorem 22
6
6.1. Theindextheorem 22
1
6.2. Morseindexofaniteration 24
0
7 6.3. Nullityofaniteration 26
0 7. EquivariantMorsetheoryfortheactionfunctional 27
/ 7.1. AbstractMorserelations 27
h
t 7.2. Homologicalinvariantsatisolatedcriticalpointsandcriticalorbits 27
a
7.3. LocalhomologicalinvariantsatcriticalO(2)-orbitsinN 29
m
7.4. EquivariantMorsetheoryforclosedgeodesics 33
:
v 8. Finalremarks e 35
i AppendixA. AnestimateontherelativehomologyoffiberbundlesoverS1 36
X
References 38
r
a
1. INTRODUCTION
ThequestionofexistenceofclosedgeodesicsisoneofthemostclassicalthemeofRie-
mannian geometry(see[31]);spectacularcontributionstothetheoryofglobalgeometry
havebeengiveninthisareabyverymanyauthors,includingHadamard,Cartan,Poincare´,
Birkhoff,Morseandmanyothers.Variationaltechniquesforestablishingtheexistenceand
themultiplicityofclosedgeodesicshavebeendevelopedandemployedbymanyauthors,
Date:January23rd,2007.
2000MathematicsSubjectClassification. 53C22,58E10,53C50,37B30.
F.M.andP.P.arepartiallysponsoredbyCNPqandFapesp.
1
2 L.BILIOTTI,F.MERCURI,ANDP.PICCIONE
including among others Ljusternik, Schnirelman, Fet, Klingenberg, Gromolland Meyer.
Closedgeodesicsarecriticalpointsofthegeodesicactionfunctionalinthespaceofclosed
paths,andexistenceresultsmaybeobtainedbyapplyingglobalvariationaltechniquesto
thisvariationalproblem.Inparticular,MorsetheoryhasbeenusedbyGromollandMeyer
(see [21]) to establish the celebratedresult on the existence of infinitely manygeometri-
cally distinct closed geodesicsin simply connectedRiemannian manifolds, whose space
offreeclosedcurveshasunboundedrationalBettinumbers.
Astotheexistenceofclosedgeodesicsinmanifoldsendowedwithanonpositivedefi-
nitemetric,veryfewresultsareavailableintheliterature,andbasicallynothingisknown
ontheirmultiplicity.AnearlierresultbyTipler(see[47])givestheexistenceofoneclosed
timelike geodesic in compact Lorentzian manifolds that admit a regular covering which
hasacompactCauchysurface.Morerecently,Guediri(see[27,28])hasextendedTipler’s
result to the case that the Cauchy surface in the coveringis not necessarily compact. In
this situation, a closed geodesic is proven to exist in each free timelike homotopy class
whichisdeterminedbya centraldecktransformation. Itisalsoprovedin[27]thatcom-
pactflatspacetimescontaina causal(i.e., nonspacelike)closedgeodesic, andin [28] the
author proves that such spacetimes contain a closed timelike geodesic if and only if the
fundamentalgroup of the underlyingmanifold contains a nontrivialtimelike translation.
TheexistenceofclosedtimelikegeodesichasbeenestablishedalsobyGallowayin[14],
wheretheauthorprovestheexistenceofa longestclosedtimelikecurve,whichisneces-
sarilyageodesic,ineachstablefreetimelikehomotopyclass. Alsononexistenceresults
fornonspacelikegeodesicsareavailable,see[15,29].
AlltheseresultsarebasedonthenotionofLorentziandistancefunction(see[5,Ch.4]).
Recall thatin Lorentziangeometryonly nonspacelikegeodesicshave lengthextremizing
properties,whileforspacelikegeodesicsusualgeometricalconstructions(curveshortening
methods)do notwork. The questionofexistence ofclosed geodesicsof arbitrarycausal
character has to be studied using the quadratic geodesic action functional in the Hilbert
manifold of closed paths of Sobolev regularity H1; its critical points are typically sad-
dlepoints. IntheLorentzian(orsemi-Riemannian)case,thevariationaltheoryassociated
to the study of the critical points of this quadratic functional is complicated by the fact
that,unliketheRiemanniancounterpart,theconditionofPalaisandSmaleisneversatis-
fied. Moreover,this functionalis notboundedfrom below, and its critical pointsalways
haveinfiniteMorseindex.In[1]theauthorsuseanapproximationschemeinthetheoryof
LjusternikandSchnirelmantodeterminetheexistenceofacriticalpointofthegeodesicac-
tionfunctionalinthespaceofclosedH1curvesinaclassofproductLorentzianmanifolds,
whosemetricisofsplittingtype. Suchcriticalpointcorrespondstoaspacelikeclosedge-
odesic;inthissituation,thankstotheresultofGalloway[14],onehastwogeometrically
distinctclosedgeodesic,oneistimelikeandtheotherisspacelike.A.Masiellohasproved
the existence of one (spacelike) closed geodesic in standard stationary Lorentzian mani-
foldsM = M ×RwhosespatialcomponentM iscompact. Morerecently(see[10]),
0 0
usingvariationalmethodstheauthorshaveestablishedtheexistenceofaclosedgeodesic
ineachfreehomotopyclasscorrespondingtoanelementofthefundamentalgrouphaving
finiteconjugacyclass, inthecaseofstaticcompactLorentzianmanifold. In[43],theau-
thorshowsthatoneclosedtimelikegeodesicexistsincompactLorentzianmanifoldsthat
areconformallystatic,providedthatthatthegroupofdecktransformationsofsomeglob-
allyhyperbolicregularcoveringofthemanifoldadmitsafiniteconjugacyclasscontaining
aclosedtimelikecurve.
In thispaper, we developa Morse theoryfor closed geodesicsin a class of stationary
Lorentzian manifolds, obtaining a result of existence of infinitely many distinct closed
geodesicsanalogoustothecorrespondingresultofGromollandMeyerintheRiemannian
case. Moreprecisely,theresultwillholdforsimplyconnectedstationarymanifoldswhose
freeloopspacehasunboundedBettinumbers(relativelytoanycoefficientfield),andthat
MORSETHEORYFORLORENTZIANCLOSEDGEODESICS 3
admitacompactCauchysurface.1Letusrecallbrieflytheessentialingredientsrequiredin
GromollandMeyer’stheory.Oneconsidersthegeodesicactionfunctionalf ontheHilbert
manifoldΛM ofallclosedpathsofSobolevclassH1onacompactandsimplyconnected
Riemannianmanifold(M,g);thisfunctionalisboundedfrombelow,itsatisfiesthePalais–
Smaleconditionanditscriticalpointsareexactlytheclosedgeodesics.Thecompactgroup
O(2)actsequivariantlyonΛM viatheoperationofO(2)ontheparametercircleS1;the
orbitsofthisactionaresmooth(compact)submanifoldsofΛM. Inparticular,thecritical
pointsoff areneverisolated;nevertheless,usingageneralizedMorseLemmaforpossibly
degenerateisolatedcriticalpoint(see[20]),generalizedMorseinequalitiescanbeapplied
toobtainestimatesonthenumberofcriticalorbits,providedthattheseorbitsareisolated.
Finally, one has to distinguish between critical orbits that correspond to iterates of the
same closed geodesic. This is done using an iteration formula for the Morse index (and
the nullity) of the n-fold covering of a given closed geodesic, which is obtained from a
celebratedresultduetoBott(see[8]). UsingaMorseindextheoremforclosedgeodesics,
theMorseindexofthen-thiteratedofaclosedgeodesicisproventobeeitherbounded,or
tohavelineargrowthinn.Usingthisfactoneprovesthatif(M,g)hasonlyafinitenumber
of geometrically distinct closed geodesics, then the rational Betti numbers of ΛM must
formaboundedsequence.Restrictiontothecaseofafieldofcharacteristiczerowasused
bytheauthorstoproveanestimateonthedimensionofrelativehomologyspacesofcertain
fiberbundles;anelementaryargumentbasedontheMayer–Vietorissequence,discussedin
AppendixA,showsthatsuchrestrictionisnotnecessary.2Thus,ifΛM hasanunbounded
sequence of Betti numbers, (M,g) must contain infinitely many geometrically distinct
closed geodesics. It is known (see [48]), that the existence of an unbounded sequence
of rational Betti numbers of the free loop space of M is equivalent to the fact that the
rational cohomologyalgebraofM isgeneratedbyatleasttwoelements. Inparticular,if
M hasthesamehomotopytypeoftheproductoftwosimplyconnectedcompactmanifold,
thenΛM hasunboundedrationalBettinumbers. Ziller[50]hasprovedthatanycompact
symmetricspaceofrankgreaterthan1hasunboundedZ -Bettinumbers;McClearyand
2
Ziller[34]havelaterprovedthatthesameconclusionholdsforcompact,simplyconnected
homogeneousspaceswhicharenotdiffeomorphictoasymmetricspaceofrankone.
Several extensions of the theory have been developed in the context of Riemannian
manifolds (see [2, 3, 4, 22, 23, 24, 45]), Finsler manifolds ([37]) and, recently, of Rie-
mannianorbifolds(see[25]).Reference[46]isagoodsurveypaperontheclassicalresults
ofGromoll–Meyertype.
When passing to the case of Lorentzianmetrics, none of the argumentsaboveworks.
First, the geodesicactionfunctionalf is notboundedfrombelow andit doesnotsatisfy
thePScondition;besides, theMorseindexofeachcriticalpointisinfinite. Inthispaper
we consider the case of stationary Lorentzian manifolds that admit a complete timelike
Killingvectorfield. Timelikeinvarianceofthemetrictensorallowstodetermineasmooth
embeddedsubmanifoldN ofΛM withthefollowingproperties:
• f| hasthesamecriticalpointsoff;
N
• N hasthesamehomotopytypeofΛM;
• f isboundedfrombelowanditsatisfiesthePSconditiononeachconnectedcom-
ponentofN;
• eachcriticalpointoff| hasfiniteMorseindex;
N
• ifacriticalpointisdegenerateforf| ,thenitisalsodegenerateforf.
N
The abeliangroupG = O(2)×R acts (isometrically)on N, and f is G-invariant. The
groupO(2)actsontheparameterspaceS1 ofthecurves,andasintheRiemanniancase,
thisactionisnotsmooth,butonlycontinuous. Nevertheless,ifγ isasmoothcurve,then
1RecallthatanytwoCauchysurfacesofagloballyhyperbolicspacetimearehomeomorphic.
2ExtensionoftheGromollandMeyerresulttononzerocharacteristicseemstohavebeenestablishedinthe
subsequentliterature.
4 L.BILIOTTI,F.MERCURI,ANDP.PICCIONE
theorbitO(2)γ issmooth,anditisdiffeomorphictoO(2)ifγ isnotconstant. Inparticu-
lar,criticalorbitsarealwayssmooth. ThegroupRactsbytranslationalongtheflowlines
ofthetimelikeKillingvectorfield;obviously,theactionsofO(2)andofRcommute. In
thissituation,wedefinegeometricallydistincttwoclosedgeodesicsthatbelongtodiffer-
entG-orbits,andthatcannotbeobtainedonefromanotherbyiteration. TheactionofR
is free, the orbitspace givenby the quotientN = N/R is a smoothmanifoldandN is
diffeomorphictotheproductN×R.Thus,inordertostudymultiplicityofdistinctclosed
geodesics,itsufficestostudygeometricallydisetinctcriticalO(2)-orbitsfortheconstrained
functional f|e. The central eresult of this paper, which gives the existence of infinitely
N
manydistinct closed geodesicsin a class of stationary Lorentzianmanifolds, is obtained
applyingequivariantMorsetheorytothissetup. Essentialtoolsforthedevelopmentofthe
theoryarea calculationof the Morse indexfor eachcriticalpointof f| , and a formula
N
thatdescribesitsgrowthunderiterations.TheMorseindexisgivenintermsofsymplectic
invariants of the geodesic, such as the Conley–Zehnder and the Maslov index, and it is
computedexplicitlyinTheorem6.4,whichisaMorseindextheoremforpossiblydegen-
erateclosedLorentziangeodesics. Thisresultis obtainedbypurelyfunctionalanalytical
techniques,provingapreliminaryresult(Theorem3.10)thatgivesamethodforcomputing
theindexofessentiallypositivesymmetricbilinearforms,possiblydegenerate,intermsof
restrictionstopossiblydegeneratesubspaces. Webelievethatthisresulthasinterestinits
own,andthatitsapplicabilityshouldgobeyondthepurposesofthepresentpaper. Using
thismethod,onereducesthecomputationoftheMorseindexforperiodicgeodesicstothe
Morseindexofthecorrespondingfixedendpointgeodesic,avoidingtheusualassumption
of orientability of the closed geodesic (see [39]). The Morse index theorem is given in
termsofasymplecticinvariantofthegeodesic,calledtheMaslovindex;inordertoesti-
mateitsgrowthbyiteration,weusearecentformulathatgivesanestimateonthegrowth
of anothersymplectic invariant, called the Conley–Zehnderindex (Proposition4.3). For
orientationpreservinggeodesics,thetwoindicesarerelatedbyasimpleformula,involving
afour-foldindex,whichiscalledtheHo¨rmanderindex(Proposition4.2).Usingthegrowth
formulafor the Conley–Zehnderindex and the (nontrivial) fact that the Morse index is,
up to a boundedperturbation,nondecreasingby iteration(Lemma6.5), we then obtaina
superlinearestimateonthegrowthoftheMaslovindexofaniterateofaclosedgeodesic
(Proposition6.7andCorollary6.8). Astothenullityofaniterate,theresultistotallyanal-
ogoustotheRiemanniancaseusingthelinearizedPoincare´map(Lemma6.9). Thissetup
paves the path to an application of infinite dimensionalequivariantMorse theory, in the
samespiritasGromollandMeyer’scelebratedresult,thatgivestheexistenceofinfinitely
manycriticalpointsforthefunctionalf|e.
N
We will now give a formal statement of the main result of the paper. Let (M,g) be
a globallyhyperbolicstationaryLorentzianmanifold,andletusassumethatM admitsa
completetimelikeKillingvectorfieldY. DenotebyF ,t∈R,theflowofY;clearly,ifγ
t
isa(closed)geodesicinM,thenalsoF ◦γisa(closed)geodesicforallt∈R.
t
In order to state our main result, we need to give an appropriate notion of geometric
equivalenceofclosedgeodesics.
Definition. Givenclosedgeodesicsγ :[a ,b ]→M,i=1,2,inastationaryLorentzian
i i i
manifold(M,g),wewillsaythattheyaregeometricallydistinct,ifthereexistsnot ∈ R
suchthatthesetsF ◦γ [a ,b ] andγ [a ,b ] coincide.
t 1 1 1 2 2 2
Themainresultofthis(cid:0)paperis(cid:1)thefoll(cid:0)owing: (cid:1)
Theorem. Let (M,g) be a simply connected globally hyperbolic stationary Lorentzian
manifold having a complete timelike Killing vector field, and having a compact Cauchy
surface. AssumethatthefreeloopspaceΛM hasunboundedBettinumberswithrespect
tosomecoefficientfield. Then,thereareinfinitelymanygeometricallydistinctnontrivial
(i.e.,nonconstant)closedgeodesicsinM.
MORSETHEORYFORLORENTZIANCLOSEDGEODESICS 5
Notethat, bycausality, everyclosedgeodesicin(M,g) isspacelike. Itshouldbeob-
servedherethat,althoughthenotionofgeometricequivalencegivenabovedependsonthe
choice of a complete timelike Killing vector field, the propertyof existence of infinitely
manygeometricallydistinctclosedgeodesicsisintrinsicto(M,g)(seeRemark8.1). Itis
alsointerestingtoobservethatthestatementoftheTheoremadmitsamildgeneralization
tothecaseofnonsimplyconnectedmanifolds(seeRemark8.4).
The paper is organized as follows. Section 2 contains a few basic facts concerning
bilinearformsandtheirindex;inSection3weprovethemainresultconcerningthecom-
putationof the indexof an essentially positivesymmetric bilinearformon a real Hilbert
space(Theorem3.10). InSection4werecallthenotionsofConley–Zehnderindexfora
continuoussymplecticpath,andofMaslovindexfora continuousLagrangianpath. The
centralresultisaninequality(Corollary4.7)thatprovidesanestimateonthegrowthofthe
MaslovindexoftheiterateofaperiodicsolutionofaHamiltoniansystem.Thedefinitionof
suchindexdependsonthechoiceofaperiodicsymplectictrivializationalongthesolution
oftheHamiltonian.Whenappliedtothecaseofperiodicgeodesicsonasemi-Riemannian
manifoldM,underacertainorientabilityassumptionwehaveacanonicalchoiceofaclass
ofperiodicsymplectictrivializationsalongthecorrespondingperiodicsolutionofthegeo-
desicHamiltonianinthecotangentbundleTM∗(Subsection4.4),andwethereforeobtain
estimatesonthegrowthoftheMaslovindexoforientationpreservingperiodicgeodesics.
TheresultsinSection4arevalidforclosedgeodesicsinarbitrarysemi-Riemannianman-
ifolds. Section5containssomematerialonthegeodesicvariationalprobleminstationary
LorentzianmanifoldsandonthePalais–Smaleconditionoftherelativeactionfunctional.
InSection6weproveageneralversionoftheMorseindextheoremforclosedgeodesics
instationaryLorentzianmanifold,thatholdsinthegeneralcaseofpossiblydegenerateand
nonorientationpreservinggeodesics(Theorem6.4). Thisisobtainedasanapplicationof
Theorem3.10, which reducesthe periodiccase to the case of fixed endpointsgeodesics.
In subsection6.2 we first show that the Morse indexof an N-th iterate is nondecreasing
onN,uptoaddingaboundedsequence. Then,weusetheMorseindextheoremandthe
estimatesonthegrowthoftheMaslovindextogetanestimateonthegrowthoftheMorse
indexunderiteration. Thecentralresult(Proposition6.7, Corollary6.8), whichprovides
analternativeapproachtotheiterationtheoryofBott[8]alsofortheRiemanniancase,says
thattheindexofanN-thiterateiseitherboundedorithaslineargrowthinN,uptoadding
aboundedsequence.ThenullityofaniterateisstudiedinSubsection6.3,andtheresultis
totallyanalogoustotheRiemanniancase. Finally,inSection7,weuseequivariantMorse
theoryforisolatedcriticalO(2)-orbitsoftheactionfunctionalf in N toproveourmain
result. WefollowcloselytheoriginalpaperbyGromollandMeyer,butwetakeadvantage
ofamorerecentapproachtoequivariantMorsetheory([12,49]),thaetsimplifiessomeof
the constructionsin [21]. The local homological invariantat an isolated critical orbit is
defined as the relative homologyof the critical sublevel, modulo the sublevel minus the
criticalorbit.Usingexcision,thisinvariantiscomputedastherelativehomologyofafiber
bundleoverthe circlemoduloa subbundle; these bundlescan be describedasassociated
bundlestotheprincipalfiberbundleO(2) → O(2)/Γ,whereΓ ⊂ SO(2)isthestabilizer
oftheorbit. OneofthecrucialstepsinGromollandMeyerconstructionisanestimateon
thedimensionofthisrelativehomology(see(7.14));thisestimateisproveninAppendixA
inthecaseofhomologywithcoefficientsinarbitraryfieldsusingtheMayer–Vietorisse-
quencein relative homology. This allows a slightgeneralizationof the originalresult in
[21],inthatnorestrictionisposedonthecharacteristicofthecoefficientfield.
In order to make the paper essentially self-contained, and to facilitate its reading, we
haveoptedto includeinthe presentversionofthe manuscriptthefullstatementofsome
results already appearing in the literature and needed in our theory. Quotations of the
originalauthorsand complete bibliographicalreferencesare given for the proofof these
results.
6 L.BILIOTTI,F.MERCURI,ANDP.PICCIONE
2. PRELIMINARIES ABOUT BILINEARFORMS ON NORMED VECTORSPACES
Inthissectionwecollectafewelementaryfactsonbilinearformsonvectorspaces. All
vectorspacesconsideredintheentiretextareassumedtobereal.Givena(normed)vector
spaceX,wewilldenotebyX∗ its(topological)dual;letB : X ×X → Rbeabilinear
formonX. TheB-orthogonalcomplementofasubspaceS ⊂X isdefinedby:
S⊥B = x∈X :B(x,y)=0,forally ∈S ;
thekernelofBisdefinedby:(cid:8) (cid:9)
Ker(B)=X⊥B = x∈X :B(x,y)=0,forally ∈X .
GivenasubspaceS ⊂X,thenKer((cid:8)B|S×S)=S∩S⊥B;if(Si)i∈I isafam(cid:9)ilyofsubspaces
ofX,then S ⊥B = S⊥B.WesaythatBisnondegenerateifKer(B)={0}.
i∈I i i∈I i
Asubspace(cid:16)SP⊂Xis(cid:17)calledisTotropicifB|S×S =0.AssumenowthatBissymmetric.We
saythatB ispositivedefinite(resp.,positivesemi-definite)ifB(x,x) > 0forallnonzero
x ∈ X (resp.,B(x,x) ≥ 0, forallx ∈ X). Similarly,wesaythatB isnegativedefinite
(resp.,negativesemi-definite)ifB(x,x) < 0forallnonzerox ∈ X (resp.,B(x,x) ≤ 0,
forallx ∈ X). A subspaceS ⊂ X iscalledpositive(resp.,negative)forB ifB| is
S×S
positivedefinite(resp.,negativedefinite). TheindexofB isthe(possiblyinfinite)natural
numberdefinedby:
n (B)=sup dim(W):W ⊂X isanegativesubspaceforB ,
−
andtheco-indexofBisde(cid:8)finedby: (cid:9)
n (B)=n (−B).
+ −
Whennotbothn (B)andn (B)areinfinite,thesignatureofB isdefinedasthediffer-
− +
encesign(B)=n (B)−n (B).
− +
Wecollectinthefollowinglemmaalltheelementaryresultsconcerningbilinearforms
onvectorspacesthatwillbeusedthroughout.
Lemma2.1. LetX beavectorspaceandletBbeasymmetricbilinearformonX.
(1) IfX =X ⊕X isaB-orthogonaldirectsumdecomposition,i.e.,B(x ,x )=0
1 2 1 2
for all x ∈ X , x ∈ X , then Ker(B) = Ker(B| )⊕Ker(B| ).
1 1 2 2 X1×X1 X2×X2
Inparticular,B isnondegenerateifandonlyifB| andB| areboth
X1×X1 X2×X2
nondegenerate.
(2) IfS ⊂X isasubspacewithX =Ker(B)⊕S,thenB| isnondegenerate.
S×S
(3) IfX = S ⊕S⊥B thenB|S×S isnondegenerate. Conversely,ifS isfinitedimen-
sionalandB|S×S isnondegeneratethenX =S⊕S⊥B.
(4) GivenanysubspaceY ⊂X,then:
n (B| )≤n (B)≤n (B| )+codim (Y),
+ Y×Y + + Y×Y X
n (B| )≤n (B)≤n (B| )+codim (Y).
− Y×Y − − Y×Y X
(5) LetX =X ⊕X beadirectsumdecompositionsuchthatBispositivedefiniteon
1 2
X andnegativesemi-definiteonX (resp.,negativedefiniteonX andpositive
1 2 1
semi-definiteonX ). Thenn (B)=dim(X )(resp.,n (B)=dim(X )).
2 + 1 − 1
(6) Assume that X = X ⊕X is a B-orthogonal direct sum decomposition such
1 2
that B is positive definite (resp., negative definite) on both X and X . Then
1 2
B is positive definite (resp., negative definite) on X. Similarly, if B is positive
semi-definite(resp.,negativesemi-definite)onbothX andX thenB ispositive
1 2
semi-definite(resp.,negativesemi-definite)onX.
(7) LetS ⊂X beamaximalsubspaceonwhichBispositivedefinite(resp.,negative
definite).ThenBisnegativesemi-definite(resp.,positivesemi-definite)onS⊥B.
MORSETHEORYFORLORENTZIANCLOSEDGEODESICS 7
(8) LetX =X ⊕X beaB-orthogonaldirectsumdecomposition.Then:
1 2
n (B)=n (B| )+n (B| ),
+ + X1×X1 + X2×X2
n (B)=n (B| )+n (B| ).
− − X1×X1 − X2×X2
(9) LetS ⊂X,N ⊂Ker(B)besubspaceswithX =N ⊕S. Then:
n (B| )=n (B), n (B| )=n (B).
+ S×S + − S×S −
(10) Let Y be a vector space and let q : X → Y be surjective linear map with
Ker(q) ⊂ Ker(B). Then there exists a unique map B : Y × Y → R such
that:
B q(x ),q(x ) =B(x ,x ), forallx ,x ∈X;
1 2 1 2 1 2
themapBis(cid:0)asymmetric(cid:1)bilinearformonY. Moreover:
(2.1) Ker(B)=q Ker(B) ∼=Ker(B)/Ker(q),
(2.2) n+(B)=n(cid:0)+(B), (cid:1)n−(B)=n−(B).
Inparticular,ifKer(q)=Ker(B)thenBisnondegenerate.
(11) Thefollowingformulaholds:
(2.3) dim(X)=n (B)+n (B)+dim Ker(B) .
+ −
(12) LetX1,X2 ⊂X beB-orthogonalsubspaces,i.e.,(cid:0)B(x1,x2(cid:1))=0forallx1 ∈X1,
x ∈X . IfX =X +X (notnecessarilyadirectsum)then:
2 2 1 2
n (B)=n (B| )+n (B| ),
+ + X1×X1 + X2×X2
n (B)=n (B| )+n (B| ).
− − X1×X1 − X2×X2
(13) Let B be a symmetric bilinearform ona vectorspace X. If B is positive semi-
definiteornegativesemi-definitethen:
Ker(B)= x∈X :B(x,x)=0 .
InparticularifB ispositives(cid:8)emi-definite(resp.,neg(cid:9)ativesemi-definite)andnon-
degeneratethenBispositivedefinite(resp.,negativedefinite).
(14) IfBisnondegenerateandsymmetric,andS ⊂X isanisotropicsubspace,then:
dim(S)≤n (B), dim(S)≤n (B). (cid:3)
− +
Let us now consider a (real) normedspace X. If T : X → Y is a continuouslinear
mapbetweennormedspacesthenT∗ : Y∗ → X∗ denotesthetransposemapdefinedby
T∗(α)=α◦T. IfS ⊂X isasubspacewedenotebySo ⊂X∗ theannihilatorofS,i.e.:
So = α∈X∗ :α| =0 .
S
IfX,Y areBanachspacesandT :(cid:8)X →Y isacontin(cid:9)uouslinearmapthenKer(T∗)=
Im(T)o and Im(T∗) ⊂ Ker(T)o. Moreover, if Im(T) is closed in Y then Im(T∗) =
Ker(T)o. Givena closedsubspaceS ⊂ X, denotebyq : X → X/S thequotientmap;
thenq∗ :(X/S)∗ →X∗ isinjectiveanditsimageequalsSo. Moreover,ifX isreflexive,
byidentifyingX withX∗∗intheusualway,thebi-annihilator(So)oequalstheclosureof
S.
Remark 2.2. If X, Y are sets and f : X → Y is a map then a subset S ⊂ X is called
saturatedfor f (orf-saturated)if x ∈ S, x ∈ X andf(x ) = f(x )imply x ∈ S.
1 2 1 2 2
If X, Y are vectorspaces, f is linear and S ⊂ X is a subspace then S is f-saturatedif
andonly if Ker(f) ⊂ S. Observethatif X, Y are Banachspacesand f : X → Y is a
surjectivecontinuouslinearmapthen,bytheopenmappingtheorem,f isaquotientmap
in the topologicalsense; hencea saturated subsetS ⊂ X is open (resp., closed)in X if
and onlyif f(S) is open(resp., closed)in Y. Similarly, a subsetU ⊂ Y is open(resp.,
closed)inY ifandonlyiff−1(U)isopen(resp.,closed)inX.
8 L.BILIOTTI,F.MERCURI,ANDP.PICCIONE
Lemma2.3. LetX beaBanachspaceandletY ⊂ X beafiniteco-dimensionalclosed
subspace.IfZ ⊂X isasubspacewithY ⊂Z thenZ isalsoclosedinX.
Proof. If q : X → X/Y denotesthe quotientmapthenq(Z) isclosed in X/Y because
X/Y isfinitedimensional. ButsinceY ⊂ Z wehavethatZ isq-saturatedandthusZ is
closedinX (seeRemark2.2). (cid:3)
Lemma2.4. LetX beaBanachspace,Y ⊂ X beaclosedsubspaceandY ⊂ X bea
1 2
finitedimensionalsubspace.ThenY +Y isclosedinX.
1 2
Proof. Ifq : X →X/Y denotesthequotientmapthenq(Y )isclosedinX/Y because
1 2 1
itisfinitedimensional.ButthenY +Y =q−1 q(Y ) isalsoclosedinX. (cid:3)
1 2 2
Finally,letusprovethefollowing: (cid:0) (cid:1)
Lemma 2.5. Let X be a reflexive Banachspace, B be a continuoussymmetric bilinear
formonXandS ⊂Xbeasubspace.IfIm(B)+SoisclosedinX∗thenthebi-orthogonal
complementofS isgivenby:
(S⊥B)⊥B =S+Ker(B).
Proof. Clearly,S⊥B =B−1(So)and:
(S⊥B)⊥B =B−1 B−1(So) o .
Denote by q : X∗ → X∗/So the quotient(cid:2)m(cid:0)ap (obser(cid:1)ve(cid:3)that an annihilator is always
closed). ObviouslyB−1(So)=Ker(q◦B). SinceIm(B)+Soisq-saturatedandclosed
inX∗,wehavethatIm(q◦B)isclosedinX∗/So(recallRemark2.2);then,wehave:
B−1(So) o =Ker(q◦B)o =Im (q◦B)∗ .
Now,usingthefactth(cid:0)at(q◦B)∗(cid:1)=B∗◦q∗,weobtain:(cid:0) (cid:1)
Im (q◦B)∗ =B(S).
Finally: (cid:0) (cid:1)
(S⊥B)⊥B =B−1 B−1(So) o =B−1 B(S) =S+Ker(B). (cid:3)
3. ON THEINDEX(cid:2)(cid:0)OF ESSENT(cid:1)IA(cid:3)LLYPOS(cid:0)ITIVE(cid:1)BILINEAR FORMS
Inthissectionwewilldiscusssomefunctionalanalyticalpreliminariesneededforthe
indextheorem.ThecentralresultisTheorem3.10,thatgivesaresultconcerningthecom-
putationoftheindexofsymmetricbilinearforms,possiblydegenerate,usingrestrictions
topossiblydegeneratesubspaces. Throughoutthissectionwewillalwaysidentifycontin-
uousbilinear formsB : X ×X → R on a normedspace X with the continuouslinear
mapB :X →X∗ givenbyx7→B(x,·).
Let(X,h·,·i)beaHilbertspaceandletB :X×X →Rbeacontinuousbilinearform.
WesaythatacontinuouslinearoperatorT :X →X representsBwithrespecttoh·,·iif:
B(x,y)= T(x),y ,
forallx,y ∈ X. Observethatifi : X → X(cid:10)∗ denote(cid:11)stheisometryx 7→ hx,·igivenby
Rieszrepresentationtheorem,thenT representsB ifandonlyifi◦T = B. Inparticular,
Ker(T) = Ker(B). A continuousbilinear form B : X ×X → R on a Banach space
X is called strongly nondegenerateif the linear map B : X → X∗ is an isomorphism.
Obviously if B is strongly nondegeneratethen B is nondegenerate. The converse holds
if we know that the linear map B : X → X∗ is a Fredholmoperatorof indexzero (for
instance,acompactperturbationofanisomorphism).
Let(X,h·,·i) bea Hilbertspace. A continuouslinearoperatorP : X → X iscalled
positive if the bilinear form hP·,·i represented by P is symmetric and positive semi-
definite.Werecallthefollowingstandardresult:
MORSETHEORYFORLORENTZIANCLOSEDGEODESICS 9
Lemma3.1. Let(X,h·,·i)beaHilbertspaceandletP :X →X beacontinuouslinear
operator. Then the bilinearform hP·,·i represented by P is an innerproducton X that
definesthesametopologyash·,·iifandonlyifP isapositiveisomorphismofX. (cid:3)
Oneproveseasilythefollowing:
Lemma3.2. LetX beaBanachspace,BbeacontinuousbilinearformonX andS ⊂X
be a closed subspace. If X = S ⊕ S⊥B then B|S×S is nondegenerate. Conversely, if
B|S×S isstronglynondegeneratethenX =S⊕S⊥B. (cid:3)
Definition 3.3. Let (X,h·,·i) be a Hilbert space and let B be a continuous symmetric
bilinearformonX. WesaythatB isessentiallypositiveiftheoperatorT : X → X that
representsB is oftheformT = P +K, with P : X → X a positiveisomorphismand
K :X →X a(symmetric)compactoperator.
Thefollowingsummarizesthemainpropertiesofessentiallypositivebilinearforms.
Lemma3.4. LetX andY be(real)Hilbertspaces. Thefollowingresultshold:
(1) ifB :X ×X →RisacontinuoussymmetricbilinearformthenBisessentially
positive if and only if there exists an inner product h·,·i on X and a compact
1
operatorK :X →X suchthatB = h·,·i +hK·,·iandsuchthath·,·i defines
1 1
thesametopologyonX ash·,·i.
(2) IfT :X →Y isacontinuousisomorphismandB :Y ×Y →Risanessentially
positive symmetric bilinear form, then the pull-back T∗B = B(T·,T·) : X ×
X →Risalsoessentiallypositive.
(3) IfB : X ×X → Risanessentiallypositivesymmetricbilinearform,thenB is
alsoessentiallypositivewithrespecttoanyotherinnerproducth·,·i onX which
1
definesthesametopologyash·,·i.
(4) If B : X ×X → R is anessentially positive symmetric bilinearform, then for
everyclosedsubspaceS ⊂X,therestrictionB| isagainessentiallypositive.
S×S
(5) if B is an essentially positive symmetric bilinear form on X, q : X → Y is
a surjective continuouslinear map with Ker(q) ⊂ Ker(B) and B is defined as
in part (10) in Lemma 2.1 then B is also (bilinear, symmetric, continuous and)
essentiallypositive.
(6) IfB : X ×X → Risanessentiallypositivesymmetricbilinearform,thenthere
existsaninnerproducth·,·i onXthatdefinesthesametopologyash·,·iandsuch
1
thatBisrepresentedwithrespecttoh·,·i byanoperatoroftheformidentityplus
1
compact.
(7) IfB :X×X →Risanessentiallypositivesymmetricbilinearform,thenKer(B)
isfinitedimensionalandn (B)isfinite.
−
Proof.
(1) ItisastraightforwardconsequenceofLemma3.1.
(2) If B is represented by P + K with P : Y → Y a positive isomorphism and
K : Y → Y compactthen the pull-backT∗B is representedby T∗(P +K)T,
whereT∗isidentifiedwithalinearmapfromY toX,usingRieszrepresentation
theorem.NowitiseasytoseethatT∗PT isapositiveisomorphismofX andthat
T∗KT isacompactoperatoronX.
(3) Followsfromitem(2),settingT =Id:(X,h·,·i )→(X,h·,·i).
1
(4) WriteB = h·,·i +hK·,·i,asinitem(1). Denotebyπ : X →S theorthogonal
1
projectionontoSandsetK′ =π◦K| . Thenh·,·i | isaninnerproductonS
S 1 S×S
thatdefinesthesametopologyash·,·i| andB| −h·,·i | isrepresented
S×S S×S 1 S×S
byK′ :S →S,whichisacompactoperator.
(5) LetS betheorthogonalcomplementofKer(q)withrespecttotheHilbertspace
innerproductofX. Thenq| :S →Y isanisomorphismandBisthepull-back
S
ofB| by(q| )−1. Theconclusionfollowsfromitems(2)and(4).
S×S S
10 L.BILIOTTI,F.MERCURI,ANDP.PICCIONE
(6) Takeh·,·i tobetheinnerproductwhoseexistenceisgiveninitem(1).
1
(7) Byitem(6),wecanchoosetheinnerproductonX suchthatB isrepresentedby
Id+K, with K compact. Then Ker(B) = Ker(Id+K) is finite-dimensional,
because Id+K is a Fredholm operator. If X ⊂ X denotes the λ-eigenspace
λ
of Id + K then by the spectral theorem for compact self-adjoint operators on
realHilbertspaces,X istheclosureofthealgebraic(h·,·i-orthogonal)directsum
X , where σ(Id+K) ⊂ R denotes the spectrum of the operator
λ∈σ(Id+K) λ
Id+K;moreover,forλ6=1,X isfinite-dimensional.Set:
λ
L
X = X , X = X ;
− λ + λ
λ∈σM(Id+K) λ∈σM(Id+K)
λ<0 λ≥0
thenitiseasytoseethatB isnegativedefiniteonX , positivesemi-definiteon
−
X and that X = X ⊕X . Finally, from part (5) in Lemma 2.1, n (B) =
+ − + −
dim(X ) < +∞, recalling that λ ∈ σ(Id+K) : λ < 0 is finite, because
−
σ(Id+K)isboundedandhas1asitsonlylimitpoint. (cid:3)
(cid:8) (cid:9)
Remark3.5. IfacontinuoussymmetricbilinearformBonaHilbertspaceXisessentially
positivethenBisnondegenerateifandonlyifBisstronglynondegenerate.Namely,Bis
representedbyaFredholmoperatorofindexzero.
Remark3.6. IfB isacontinuoussymmetricbilinearformonaHilbertspaceX whichis
essentiallypositiveandifS ⊂X isasubspacethen:
(S⊥B)⊥B =S+Ker(B);
namely,we onlyhaveto check the hypothesesofLemma2.5. ObviouslyX is reflexive,
beingaHilbertspace.Moreover,Im(B)isclosedandfiniteco-dimensionalinX∗,because
BisaFredholmoperator;thus,Im(B)+SoisclosedinX∗,byLemma2.3.
Lemma3.7. LetX beaHilbertspaceandletBbeacontinuoussymmetricbilinearform
onX whichisessentiallypositive. IfW ⊂X isaclosedsubspacethenW +W⊥B isalso
closedinX.
Proof. Wecanchoosetheinnerproducth·,·ionX suchthatBisrepresentedbyanoper-
atoroftheformT = Id+K,withK compact(seeLemma3.4,item(6)). IfW′ denotes
theorthogonalcomplementofW withrespecttoh·,·i thenW⊥B = T−1(W′). We then
have to show that W + T−1(W′) is closed in X. Since T is a Fredholm operator, its
image is closed in X and so T : X → Im(T) is a surjective continuous linear opera-
tor between Banach spaces. We have Ker(T) ⊂ T−1(W′) ⊂ W +T−1(W′), so that
W +T−1(W′)isT-saturated(seeRemark2.2);thusW +T−1(W′)isclosedinX ifand
onlyifT W +T−1(W′)]=T(W)+ W′∩Im(T) isclosedinIm(T). But:
(cid:2) T(W)+ W′∩Im(T(cid:0)) = T(W)(cid:1)+W′ ∩Im(T),
andthereforetheproofwill(cid:0)becompleted(cid:1)once(cid:0)weshowthat(cid:1)T(W)+W′ isclosedinX.
Wehave:
T(W)+W′ = x+y+K(x):x∈W, y ∈W′ =Im(Id+K◦π),
whereπ denotestheorth(cid:8)ogonalprojection(withrespectto(cid:9)h·,·i)ontoW. SinceK ◦π is
compact,Id+K◦πisaFredholmoperatorandhenceitsimageisclosedinX. (cid:3)
Lemma3.8. LetXbeaHilbertspaceandletBbeanondegeneratecontinuoussymmetric
bilinearformonX whichisessentiallypositive. IfZ ⊂X isanisotropicsubspacethen:
(3.1) n−(B)=n−(B|Z⊥B×Z⊥B)+dim(Z),
allthetermsontheequalityabovebeingfinitenaturalnumbers.