morse functions and their gradients

《morse functions and their gradients》由会员上传分享，免费在线阅读，更多相关内容在工程资料-天天文库。

1、CHAPTER2MorsefunctionsandtheirgradientsAcriticalpointpofaC∞functionfonamanifoldiscallednon-degenerateifthebilinearformf(p):Tp(M)×Tp(M)→Risnon-degenerate.Theindexofthisformiscalledtheindexofp.AMorsefunctiononamanifoldisaC∞function,suchthatitscriticalpoin

2、tsareallnon-degenerate.WebeginwiththeclassicalMorselemma,whichsaysthateveryMorsefunctioninaneighbourhoodofitscriticalpointofindexkisdiffeomorphictothefunctionQk+const,whereQkisaquadraticformofindexk.WeprovethenthatthesubsetofallMorsefunctionsonaclosedman-i

3、foldisopenanddenseinthesetofallC∞functionsonthemanifold(Theorem1.30;thisresultisdeducedfromamoregeneralTheorem1.25).InthesecondsectionweintroducethegradientsofMorsefunctionsandforms.Recallthatthegradientofadifferentiablefunctionf:Rm→Risthevectorfield∂f∂fg

4、radf(x)=(x),...,(x).∂x1∂xmThenotionofgradientgeneralizesimmediatelytosmoothfunctionsonRiemannianmanifolds.Forsuchafunctionf:M→Rthevectorfieldgradfisdefinedbytheformulagradf(x),h=f(x)(h)(wherex∈M,h∈TxMand·,·standsforthescalarproductinducedbytheRiemannia

5、nmetriconTxM).ThisvectorfieldwillbecalledtheRiemanniangradientoff.Thefunctionfisstrictlyincreasingalonganynon-constantintegralcurveγofgradf,since(f◦γ)(t)=f(γ(t))(γ(t))=

6、

7、gradf(γ(t))

8、

9、2.Thusthepropertiesoffandtheflowgeneratedbygradfarecloselyrelatedtoeach

10、other.Ingeneralonecanusefunctionsincreasingalongeachtrajectoryofagivenvectorfieldvtostudythedynamicsoftheflowgeneratedbyv.ThisapproachwasdeeplyexploredbyA.M.Liapounov(seehisthesisdefendedin1892,andtranslatedintoFrenchin[88]).34Chapter2.MorsefunctionsInMorse

11、theorythenotionofgradientdescentwasusedalreadyintheseminalarticle[98]ofM.Morse.AveryconvenientclassofgradientflowswasintroducedandextensivelyusedbyJ.Milnorinhisbook[92]:Definition0.1([92],§3).LetMbeamanifold,f:M→RbeaMorsefunction.Avectorfieldviscalledagradie

12、nt-likevectorfieldforf,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