黎曼几何习题集-Do-Carmo

RIEMANNIANGEOMETRYPRC.ZZJToProfessorZhuForbetterunderstandingonLobatchevskiGeometry...ProblemSetRiemannianGeometryManfredoPerdigeaodoCarmoChapter0DifferentiableManifolds...................................................1,9Chapter1RiemannianMetrics...................................................1,(4),5Chapter2AffineceConnections;RiemannianConnections.........................2,3,(8)Chapter3Geodesics;ConvexNeighborhoods.........................................7,9Chapter4Curvature................................................................7,8Chapter5JacobiFields.............................................................3,4Chapter6IsometricImmersions....................................................3,11Chapter7CompleteManifolds;Hopf-RinowandHadamardTheorems............6,7,(10)Chapter8SpacesofConstantCurvature.........................................1,4,(5)Chapter9VariationsofEnergy...............................................1,(2,3),4,5Chapter10TheRauchComparisonTheorem.........................................3,5Chapter11TheMorseIndexTheorem........................................(2),(4),5,6Chapter12TheFundamentalGroupofManifoldsofNegativeCurvature............noexChapter13TheSphereTheorem..................................................noexConcludingRemarks...................................................................0DifferentiableManifolds0.1(ProductManifold).LetMandNbedifferentiablemanifoldsandlet{(Uα,xβ)},{(Vβ,yβ)}differentiablestructuresonMandN,respectively.ConsiderthecartesianproductM×Nandthemappingzαβ(p,q)=(xα(p),yβ(q)),p∈Uα,q∈Vβa)Provethat(Uα×Vβ,zαβ)isadifferentiablestructureonM×Ninwhichtheprojectionsπ1:M×N→Mandπ2:M×N→Naredif-ferentiable.WiththisdifferentiablestructureM×NiscalledtheproductmanifoldofMwithN.b)ShowthattheproductmanifoldS1×···×S1ofncirclesS1,whereS1⊂R2hastheusualdifferentiablestructure,isdiffeomorphictothen−torusTnofexample4.9a).Proof.a)Clearly,zαβ:Uα×Vβ→xα(Uα)×yβ(Vβ)⊂M×N(p,q)7→(xα(p),yβ(q))12PRC.ZZJisinjective.Moreover,[α,βzαβ(Uα×Vβ)=[αxα(Uα)×[βyβ(Vβ)=M×Nandifzαβ(Uα×Vβ)∩zγδ(Uγ×Vδ)=W6=∅thenz−1γδ◦zαβ(p,q)=z−1γδ(xα(p),yβ(q))=(x−1γ◦xα(p),y−1δ◦yβ(q))isdifferentiable.Thus,bydefinition,withthisdifferentiablestructure,M×Nisadifferentiablemanifold.b)RecallTn=Rn/Zn.LetF:S1×···×S1→Tn(eiαj)nj=17→αj2π+njnj=1whereαj∈[0,2π),nj∈ZWehave•Fisinjective,sinceαj2π+nj=βj2π+mj⇒αj−βj=2π(mj−nj)⇒eiαj=eiβj•Fissurjective,justnotethatαj∈[0,2π)⇒αj2π∈[0,1)•FandF−1aredifferentiable,thisisprovedbyalistofgraphs.Indeed,one”y−1◦F◦x”isoftheformf(t)=arctantπ−140.9LetG×M→MbeaproperlydiscontinuousactionofagroupGonadifferentiablemanifoldM.a)ProvethatthemanifoldM/G(Example4.8)isorientedifandonlyifthereexistsanorientationofMthatispreservedbyallthediffeo-morphismsofG.b)Usea)toshowthattheprojectiveplaneP2(R),theKleinbottleandtheMobiusbandarenon-orientable.c)ProvethatP2(R)isorientableifandonlyifnisodd.Proof.a)ifpart:Let(Uα,xα)beanorientationofMthatispreservedbyallthediffeomorphismsofG,i.e.W=Uβ∩g(Uα)6=∅⇒det(x−1β◦g◦xα)0RIEMANNIANGEOMETRY3Weclaimthat(π(Uα),π◦xα)isanorientationofM/G.Indeed,π(Uα)∩π(Uβ)6=∅⇒det((π◦xβ)−1◦(π◦xα))=det(x−1β◦g◦xα)0forsomeg∈G.Onlyifpart:WeknowtheatlasofM/GisinducedfromM,hencetheconclusionfollowsfromthereverseofthe”ifpart”.b)LetG={Id,A}whereAistheantipodalmap.RecallthatProjective2−spaceP2(R)=S2/G,whereS2=2−dimsphereKleinbottleK=T2/G,whereT2=2−dimtorusMobiusbandM=C/G,whereC=2−dimcylinderClearly,S2,T2,Careorientable2−dimmanifols,butAreversetheorientationofR3,henceS2,T2,C.Theconclusionfollowsfroma).c)We’vethefollowingequivalence:Pn(R)isorientable⇔ApreservestheorientationofSn(bya))⇔ApreservestheorientationofRn+1(TheorientationisinducedfromRn+1)⇔(n+1)iseven⇔nisodd1RiemannianMetrics1.1ProvethattheantipodalmappingA:Sn→SngivenbyA(p)=−pisanisometryofSn.UsethisfacttointroduceaRiemannianmetricontherealprojectivespacePn(R)suchthatthenaturalprojectionπ:Sn→Pn(R)isalocalisometry.Proof.a)AisanisometryofSn.WefirstclaimthatTpSn=TA(p)Sn.ItisenoughtoproveTpSn⊂TA(p)Sn,sinceTA(p)Sn⊂⊂TA◦A(p)Sn=TpSnIndeed,foranyv∈TpSn,∃c:(−ε,ε)→Snsuchthatc(0)=p,c0(0)=v.ThusA◦c:(−ε,ε)→SnisacurvewithA◦c(0)=A(p),(A◦c)0(0)=dAp(c0(0))=−c0(0)=−v.Hence−v∈TA(p)Snandv∈TA(p)SnsinceTA(p)Snisalinearspace.NowthefactAisanisometryofSnisclear.dAp(v),dAp(w)A(p)=−v,−w−p=v,w−p=v,wpb)ConstructionofametriconPn(R)suchthatπisalocalisometry.Foranyp∈Sn,π(p)∈Pn(R),define(dπ)p(v),(dπ)p(w)π(p),v,wpIndeed,4PRC.ZZJ•BecauseofsurjectivityofπandtheconstructionofatlasonPn(R),thevector”on”Pn(R)isoftheform(dπ)p(v),p∈Sn,v∈Tp(Sn).•Itiswell-defined.Indeed,(dπ)pissurjective,thusinjective,hencetheone-to-onecorrespondencebetween(dπ)p(v)andv.Andifπ(p)=π(q),thenq=porq=A(p).Inthelattercase,(dπ)p(v)=(d(π◦A))p(v)=(dπ)A(p)◦(dA)p(v)=(dπ)A(p)(−v)(dπ)p(w)=(dπ)A(p)(−w)−v,−wA(p)=v,wp•SincetheactionofGonMisproperlycontinuous,bydefinition,πisalocalisometry.1.4Afunctiong:R→Rgivenbyg(t)=yt+x,t,x,y∈R,y0,iscalledaproperaffinefunction.ThesubsetofallsuchfunctionwithrespecttotheusualcompositionlawformsaLiegroupG.AsadifferentiablemanifoldGissimplytheupperhalf-plane{(x,y)∈R2;y0}withthedifferentiablest