Rigidity theorems of Clifford Torus

Abstract

Let M be an n-dimensional closed minimally immersed hypersurface in the unit sphere S n+1 . Assume in addition that M has constant scalar curvature or constant Gauss-Kronecker curvature. In this note we announce that if M has (n − 1) principal curvatures with the same sign everywhere, then M is isometric to a Clifford Torus S 1 1 n × S n−1 n−1 n .