comparison theorem in riemannian geometry.pdf
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comparison theorem in riemannian geometry.pdf

comparisontheoreminriemanniangeometry.pdf

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ComparisonTheoremsinRiemannianGeometryJ.-H.Eschenburg0.IntroductionThesubjectoftheselecturenotesiscomparisontheoryinRiemanniangeometry:WhatcanbesaidaboutacompleteRiemannianmanifoldwhen(mainlylower)boundsforthesectionalorRiccicurvaturearegiven?StartingfromthecomparisontheoryfortheRiccatiODEwhichdescribestheevolutionoftheprincipalcurvaturesofequidis-tanthypersurfaces,wediscusstheglobalestimatesforvolumeandlengthgivenbyBishop-GromovandToponogov.AnapplicationisGromov'sestimateofthenumberofgeneratorsofthefundamentalgroupandtheBettinumberswhenlowercurvatureboundsaregiven.Usingconvexityarguments,weprovethe"soultheorem"ofCheegerandGromollandthespheretheoremofBergerandKlingenbergfornonnegativecur-vature.IflowerRiccicurvatureboundsaregivenweexploitsubharmonicityinsteadofconvexityandshowtherigiditytheoremsofMyers-ChengandthesplittingtheoremofCheegerandGromoll.TheBishop-GromovinequalityshowspolynomialgrowthoffinitelygeneratedsubgroupsofthefundamentalgroupofaspacewithnonnegativeRiccicurvature(Milnor).WealsodiscussbrieflyBochner'smethod.Theleadingprincipleofthewholeexpositionistheuseofconvexitymethods.Fiveideasmakethesemethodswork:ThecomparisontheoryfortheRiccatiODE,whichprobablygoesbacktoL.Green[15]andwhichwasusedmoresystematicallybyGromov[20],thetriangleinequalityfortheRiemanniandistance,themethodofsupportfunctionbyGreeneandWu[16],[17],[34],themaximumprincipleofE.Hopf,generalizedbyE.Calabi[23],[4],andtheideaofcriticalpointsofthedistancefunctionwhichwasfirstusedbyGroveandShiohama[21].Wehavetriedtopresenttheideascompletelywithoutbeingtootechnical.ThesenotesarebasedonacoursewhichIgaveattheUniversityofTrentoinMarch1994.ItisapleasuretothankElisabettaOssannaandStefanoBonaccorsiwhohaveworkedoutandtypedpartoftheselectures.WealsothankEviSamiouandRobertBockformanyvaluablecorrections.Augsburg,September1994J.-H.Eschenburg11.Covariantderivativeandcurvature.Notation:ByMwealwaysdenoteasmoothmanifoldofdimensionn.ForpM,2thetangentspaceatpisdenotedbyTpM,