An optimal rigidity theorem for complete submanifolds in a sphere

It is proved that if M^n is an n-dimensional complete submanifold with parallel mean curvature vector and flat normal bundle in S^n＋p（1）, and if supM S 〈 α（n, H）, where α（n,H）=n＋n^3/2（n-1）H^2-n（n-2）/n（n-1）√n^2H^4＋4（n-1）H^2,then M^n must be the totally urnbilical sphere S^n（1/√1＋H^2）.An example to show that the pinching constant α（n, H） appears optimal is given.     

正曲率子流形的几何刚性定理

§1Introduction LetMnbeann-dimensionalcompactRiemannianmanifoldisometricallyimmersedinto an(n+p)-dimentionalcompleteandsimplyconnectedRiemannianmanifoldFn+p(c)with constantcurvaturec.DenotebyKMandHthesectionalcurvatureandmeancurvatureofM respectively.In[10],Yauprovedthefollowingstrikingresult.TheoremA.LetMnbeann-dimensionalorientedcompactminimalsubmanifoldin Sn+p(1).IfthesectionalcurvatureofMisnotlessthanp-12p-1,thenMiseitherthetotally geodesicsphere,thestandardimmersionoftheproductoftw…     

Rigidity of compact minimal submanifolds in a unit sphere

LetM be ann-dimensional compact minimal submanifold of a unit sphereS ^n+p (p2); and letS be a square of the length of the second fundamental form. IfS2/3n everywhere onM, thenM must be totally geodesic or a Veronese surface.     

On the curvature of minimal submanifolds in a sphere

S. T. Yau proved inAmer. J. Math. 97 (1975), p. 95, Theorem 15 that if the sectional curvature of ann-dimensional compact minimal submanifold in the (n + p)-dimensional unit sphere is everywhere greater than (p – 1)/(2p – 1), then this minimal submanifold is totally geodesic. In this note we improve this bound for the casep 2 to (3p – 2)/(6p).