A M(?)bius Rigidity Theorem For Hypersurfaces In Sphere Space

Let x:Mm→Sm+1(m>3)be an m-dimensional no umbilical immersed hypersurface in m+1-dimensional unit sphere Sm+1.B is the Mobius second basic form,by using the nonnegativity of modulus square of curvature tensor,we obtained a Mobius rigidity theorem,We have inequality:trB4 ≤((m-1)(m2-3m+3))/m3,the equal sign holds if and only if hypersurfaces x(Mm)is Enveloping of family of one parameter sphere.In addition,the hypersurfaces are classified under the condition that the second basic form can be decomposed.