comparison theorem in riemannian geometry

ComparisonTheoremsinRiemannianGeometryJ.-H.Eschenburg0.IntroductionThesubjectoftheselecturenotesiscomparisontheoryinRiemanniangeometry:WhatcanbesaidaboutacompleteRiemannianmanifoldwhen(mainlylower)boundsforthesectionalorRiccicurvaturearegiven?StartingfromthecomparisontheoryfortheRiccatiODEwhichdescribestheevolutionoftheprincipalcurvaturesofequidis-tanthypersurfaces,wediscusstheglobalestimatesforvolumeandlengthgivenbyBishop-GromovandToponogov.AnapplicationisGromov'sestimateofthenumberofgeneratorsofthefundamentalgroupandtheBettinumberswhenlowercurvatureboundsaregiven.Usingconvexityarguments,weprovethesoultheoremofCheegerandGromollandthespheretheoremofBergerandKlingenbergfornonnegativecur-vature.IflowerRiccicurvatureboundsaregivenweexploitsubharmonicityinsteadofconvexityandshowtherigiditytheoremsofMyers-ChengandthesplittingtheoremofCheegerandGromoll.TheBishop-GromovinequalityshowspolynomialgrowthofnitelygeneratedsubgroupsofthefundamentalgroupofaspacewithnonnegativeRiccicurvature(Milnor).WealsodiscussbrieyBochner'smethod.Theleadingprincipleofthewholeexpositionistheuseofconvexitymethods.Fiveideasmakethesemethodswork:ThecomparisontheoryfortheRiccatiODE,whichprobablygoesbacktoL.Green[15]andwhichwasusedmoresystematicallybyGromov[20],thetriangleinequalityfortheRiemanniandistance,themethodofsupportfunctionbyGreeneandWu[16],[17],[34],themaximumprincipleofE.Hopf,generalizedbyE.Calabi[23],[4],andtheideaofcriticalpointsofthedistancefunctionwhichwasrstusedbyGroveandShiohama[21].Wehavetriedtopresenttheideascompletelywithoutbeingtootechnical.ThesenotesarebasedonacoursewhichIgaveattheUniversityofTrentoinMarch1994.ItisapleasuretothankElisabettaOssannaandStefanoBonaccorsiwhohaveworkedoutandtypedpartoftheselectures.WealsothankEviSamiouandRobertBockformanyvaluablecorrections.Augsburg,September1994J.-H.Eschenburg11.Covariantderivativeandcurvature.Notation:ByMwealwaysdenoteasmoothmanifoldofdimensionn.Forp2M,thetangentspaceatpisdenotedbyTpM,andTMdenotesthetangentbundle.IfM0isanothermanifoldandf:M!M0asmooth(i.e.C1)map,itsdierentialatsomepointp2Misalwaysdenotedbydfp:TpM!Tf(p)M0.Forv2TpMwewritedfp(v)=dfp:v=@vf.Ifc:I!Misa(smooth)curve,wedenoteitstangentvectorbyc0(t)=dc(t)=dt=dct:12Tc(t)M(where12TtI=IR).Iff:M!IR,thendfp2(TpM).IfMisaRiemannianmanifold,i.e.thereexistsascalarproduct;onanytangentspaceofM,thisgivesanisomorphismbetweenTpMand(TpM);thevectorrf(p)correspondingtodfpiscalledthegradientoff.LetMbeaRiemannianmanifold.Wedenoteby;thescalarproductonMandwedenethenormofavectorbykvk=pv;v;thelengthofacurvec:I!MbyL(c)=ZIkc0(t)kdt;andthedistancebetweenx;y2Mbyjx;yj=inffL(c);c:x!yg:wherec:x!ymeansthatc:[a;b]!Mwithc(a)=xandc(b)=y.IfL(c)=jx;yjforsomec:x!y,thenciscalledshortest.TheopenandclosedmetricballsaredenotedbyBr(p)andDr(p),i.e.Br(p)=fx2M;jx;pjrg;Dr(p)=fx2M;jx;pjrg:Similarly,wedeneBr(A)foranyclosedsubsetAM.WedenotebyX(M)thesetofvectoreldsonM.Denition1.1TheLevi-CivitacovariantderivativeD:X(M)X(M)!X(M)(X;Y)!DXY;isdeterminedbythefollowingpropertiesholdingforallfunctionsf;g2C1(M)andforallvectoreldsX;X0;Y;Y02X(M):21.D(fX+gX0)Y=fDXY+gDX0Y;2.DX(fY+gY0)=(@Xf)Y+fDXY+(@Xg)Y0+gDXY0;3.DXYDYX=[X;Y]=Liebracket;4.@ZX;Y=DZX;Y+X;DZY.Denition1.2TheRiemanniancurvaturetensor(X;Y;Z)7!R(X;Y)Zisdenedasfollows:R(X;Y)Z=DXDYZDYDXZD[X;Y]ZItsatisescertainalgebraicidentities(curvatureidentities),namelyR(X;Y)Z;W=R(Y;X)Z;W=R(X;Y)W;Z=R(Z;W)X;YandtheBianchiidentityR(X;Y)Z+R(Y;Z)X+R(Z;X)Y=0(cf.[29]).Inparticular,RV:=R(:;V)VisaselfadjointendomorphismofTMforanyvectoreldVonM.Severalnotionsofcurvaturearederivedfromthistensor:1.SectionalcurvatureK(;):ForeverylinearlyindependentpairofvectorsX;Y2TpM,K(X;Y)=R(X;Y)Y;XkXk2kYk2X;Y2:KisdenedonthespaceoftwodimensionallinearsubspacesofTpM(dependingonlyonspan(X;Y)).2.RiccicurvatureRic(X;Y)=trace(Z7!R(Z;X)Y):Bythecurvatureidentities,Ric(X;Y)=Ric(Y;X).RiccicurvatureindirectionXisgivenbyRic(X)=Ric(X;X)whereXisaunitvector.3.Scalarcurvatures=trace(Ric)=XRic(Ei;Ei)3wherefEigni=1isalocalorthonormalbasis.ThereisacloserelationshipbetweenRV=R(:;V)Vandthesectionalcurvature:LetkVk=1.ForXorthogonaltoVwehaveRVX;X=R(X;V)V;X=K(V;X)kXk2Hencethehighest(+)andlowest()eigenvaluesofRVgiveaboundtoK(V;X),since(RV)RVX;XX;X+(RV):Moreover,trace(RV)=Ric(V;V).Letuscomebacktothecovariantderivative.Itiseasytoseethatforanyp2M,(DXY)pdependsonlyondYp:X(p)wherethevectoreldYisconsideredasasmoothmapY:M!TM.Therefore,thecovariantderivativeisalsodenedifthevectoreldsXandYareonlypartiallydened.E.g.if:I!MisasmoothregularcurveandYisavectoreldalong,i.e.asmoothmapY:I!TMwithY(t)2T(t)Mforallt2I(e.g.0issuchavectoreld),thenY0(t):=DY(t)dt:=D0(t)Yisdened(justextend0andYarbitrarlyoutside).Similar,if:I1:::Ik!Mdependsonkvariables,wehavekpartialderivatives@@tiandcorrespondingcovariantderivativesD@ti(i=1;:::;k)along.(Formally,avectoreldalongisasectionofthepull-backbundleTM,andDinducesacovariantderivativeonthisbundle.)Denition1.3AvectoreldYalongacurve:I!MiscalledparallelifY0=0.AcurveisacalledageodesicinMif0isparallel,i.