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1、CHAPTER2MorsefunctionsandtheirgradientsAcriticalpointpofaC∞functionfonamanifoldiscallednon-degenerateifthebilinearformf(p):Tp(M)×Tp(M)→Risnon-degenerate.Theindexofthisformiscalledtheindexofp.AMorsefunctiononamanifoldisaC∞function,suchthatitscriticalpoin
2、tsareallnon-degenerate.WebeginwiththeclassicalMorselemma,whichsaysthateveryMorsefunctioninaneighbourhoodofitscriticalpointofindexkisdi?eomorphictothefunctionQk+const,whereQkisaquadraticformofindexk.WeprovethenthatthesubsetofallMorsefunctionsonaclosedman-i
3、foldisopenanddenseinthesetofallC∞functionsonthemanifold(Theorem1.30;thisresultisdeducedfromamoregeneralTheorem1.25).InthesecondsectionweintroducethegradientsofMorsefunctionsandforms.Recallthatthegradientofadi?erentiablefunctionf:Rm→Risthevector?eld?f?fg
4、radf(x)=(x),...,(x).?x1?xmThenotionofgradientgeneralizesimmediatelytosmoothfunctionsonRiemannianmanifolds.Forsuchafunctionf:M→Rthevector?eldgradfisde?nedbytheformulagradf(x),h=f(x)(h)(wherex∈M,h∈TxMand·,·standsforthescalarproductinducedbytheRiemannia
5、nmetriconTxM).Thisvector?eldwillbecalledtheRiemanniangradientoff.Thefunctionfisstrictlyincreasingalonganynon-constantintegralcurveγofgradf,since(f?γ)(t)=f(γ(t))(γ(t))=
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7、gradf(γ(t))
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9、2.Thusthepropertiesoffandthe?owgeneratedbygradfarecloselyrelatedtoeach
10、other.Ingeneralonecanusefunctionsincreasingalongeachtrajectoryofagivenvector?eldvtostudythedynamicsofthe?owgeneratedbyv.ThisapproachwasdeeplyexploredbyA.M.Liapounov(seehisthesisdefendedin1892,andtranslatedintoFrenchin[88]).34Chapter2.MorsefunctionsInMorse
11、theorythenotionofgradientdescentwasusedalreadyintheseminalarticle[98]ofM.Morse.Averyconvenientclassofgradient?owswasintroducedandextensivelyusedbyJ.Milnorinhisbook[92]:De?nition0.1([92],§3).LetMbeamanifold,f:M→RbeaMorsefunction.Avector?eldviscalledagradie
12、nt-likevector?eldforf,if1)foreverynon-criticalpointxoffwehave:f(x)(v(x))>0,2)foreverycriticalpointpoffthereisachartΨ:U→V?RmforM,suchthat(F)f?Ψ?1(x,...,x)=f(p)?(x2+···+x2)+(x2+···+x2),1m1kk+1m(G)Ψ?(v)(x1,...,xm)=(?x
