Milnor - Morse Theory - clarified PDF.pdf

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1、MORSETHEORYBYJ.MilnorBasedonlecturenotesbyM.SPIVAKandR.WELLSPRINCETON,NEWJERSEYPRINCETONUNIVERSITYPRESS1963CONTENTSCopyright01963.byPrincetonUniversityPressAllRightsReservedPARTI.NON-DEGENERATESMOOTHFUNCTIONSONAMANIFOLDL.C.Card63-13729$1.Introduction.......................1$2.Definit

2、ionsandLemmas..................4$3.HomotopyTypeinTermsofCriticalValues........12$4.Examples.........................25$5.TheMorseInequalities..................2806.ManifoldsinEuclideanSpace:TheExistenceofNon-degenerateFunctions.................12$7.ThekfschetzTheoremonHyperplaneSecti

3、ons.......39PARTI1.AWIDCOURSEINRIEZ"NIANGEOKETRY$8.CovariantDifferentiation.............43$9.TheCurvatureTensor................51$10.GeodesicsandCompleteness.............55PARTI11.THECALCULUSOFVARIATIONSAPPLIEDTOGEODESICS$11.ThePathSpaceofaSmoothManifold...........67§12.TheEnergyofaP

4、ath...................70$13.TheHessianoftheEnergyFunctionataCriticalPath..74$14.JacobiFields:TheNull-spaceofE*++..........77515.TheIndexTheorem....................a3§16.AFiniteDimensionalApproximationtoQC........aa817.TheTopologyofthePullPathSpace...........93$18.EXistenceofNon-conju

5、gatePoints............9aPrintedintheUnitedStatesofAmerica§I9.SomeRelationsBetweenTopologyandCurvature......looVCONTENTSPARTIV.APPLICATIONSTOLIEGROUPSIICSPACES....................$20.SymmetricSpaces109.............$21.LieGroupsasSymmetricSpaces112..........$22.WholeManifoldsofMinimalG

6、eodesics118124$23.TheBottPeriodicityTheoremfortheUnitaryGroup...$24.ThePerjodicityTheoremfortheOrthogonalGroup....13’1PARTI........149APPmIX.THEHGMOTOPYTYPEOFAMONOTONEUNIONNON-DEGENERATESMOOTHFUNCTIONSONAMANIFOLD.$1.Introduction.Inthissectionwewillillustratebyaspecificexamplethesitu-

7、ationthatwewillinvestigatelaterforarbitrarymanifolds.Letuscon-sideratorusM,tangenttotheplaneV,asindicatedinDiagram1.Diagram1.Letf:M+R(Ralwaysdenotestherealnumbers)betheheightabovetheVplane,andletI@‘bethesetofallpointsxEMsuchthatf(x)5a.Thenthefollowingthingsaretrue:(1)Ifa<0

8、&isvacuous.(