Submanifolds of product Riemannian manifold.pdf

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1、2000,20B(2):213-218SUBMANIFOLDSOFPRODUCTRIEMANNIANMANIFOLD1XuSenlin()NiYilong(
)DepartmentofMathematics,UniversityofScienceandTechnologyofChina,Heifei230026,ChinaAbstractThispaperdiscussessubmanifoldsofproductRiemannianmanifold,andprovesthataninvariantsubmanifoldo

2、fproductRiemannianmanifoldcanbewrittenasapro-duction.KeywordsProductmanifold,isometric1991MRSubjectClassification53C401IntroductionLetUandVbeRiemannianmanifolds,withthedimensionn1andn2respectively.U×VistheRiemannianproductofUandV.WedenotebyPandQtheprojectionmappingsofT(

3、U×V)toTUandTVrespectively.Thenwehave22P+Q=I,P=P,Q=Q,PQ=QP=0.2WeputJ=P−Q.ItiseasytoseethatJ=I.WedefineaRiemannianmetricofU×Vbyg(X,Y)=g1(PX,PY)+g2(QX,QY),whereg1andg2areRiemannianmetricofUandVrespectively.Itfollowsthatg(JX,Y)=g(X,JY).By∇wedenotetheg’sLevi-Civitaconnection

4、.Thenwecaneasilyseethat∇J=0.(1)Infact,Fromthedefinitionofg,wecangetthatUandVareallgeodesicsubmanifoldsofU×V.∀X,Y,Z∈T(U×V),∇PX=∇PX+∇PX.(2)YPYQY∇X=∇PX+∇QX+∇QX+∇PX.YPYQYPYQYAlsobecause,DEDE∇PX,QZ=∇PX,QZ−PX,∇QZQYQYQYDE=0−PX,∇QZ=0,QY1ReceivedJun.1,1998.ProjectsupportedbyNNSF

5、CandNECYSFC.214ACTAMATHEMATICASCIENTIAVol.20Ser.Bwehave,P∇PX=∇PX,P∇QX=0.So,QYQYPYP∇X=∇PX+∇PX.(3)YPYQY
From(2)and(3),∇PX=∇PX−P∇X=0.BecauseX,Yarearbitraryvectors,YYY∇P=0.Forthesamereason∇Q=0.Sowehave∇J=∇P−∇Q=0.Now,letMbeasubmanifoldofU×V,andBthedifferentialoftheimbedding

6、iofMintoU×V,i.e.B=i∗.LetXbeatangentvectorfieldofM.ThenwecanwriteJBXinthefollowingway:⊤⊥JBX=(JBX)+(JBX)=BfX+N,wheref:TM→TMisalineartransformationofTM,andN=(JBX)⊥∈(TM)⊥.Missaidtobeaninvariantsubmanifold,ifJBX=(JBX)⊤=BfXalwaysholds.Intherestofthispaperweassumethatthesubman

7、ifoldMisinvariant.Thenwehavef2=I.For∀N∈(TM)⊥,∀X∈TMwehaveg(JN,BX)=g(N,JBX)=g(N,BfX)=0,whichimplythatJN∈(TM)⊥.Applying∇tobothsidesofJBX=BfX,weobtainBY∇BYJBX=∇BYBfX.MakinguseofGuassequation∇BYBX=B∇YX+h(Y,X),weget∇BYBfX=B∇YfX+h(Y,fX),wherehisthesecondfundamentalformofM.Ont

8、heotherhand,substituting(1),wehave∇BYBfX=∇BYJBX=(∇BYJ)BX+J∇BYBX=0+J∇BYBX=J(B∇YX+h(Y,X))=Bf∇YX+Jh(X,Y)=B∇YfX−B(∇Yf)X+J