On a characterization of round spheres

Abstract

It is shown that, an immersion of n-dimensional compact manifold withoutboundary into left( n+1right) -dimensional Euclidean space, hyperbolicspace or the open half spheres, is a totally umbilic immersion if for some %r, r=2,3,cdots,n the r-th mean curvature H_{r} does notvanish and there are nonnegative constantsC_{1},C_{2},cdots,C_{r} such thatbegin{equation*}H_{r}=sum_{i=1}^{r-1}C_{i}H_{i}.end{equation*}