Let M be an n-dimensional closed minimally immersed hypersurface in the uni
t sphere Sn+1. Assume in addition that M has constant scalar curvature or c
onstant Gauss-Kronecker curvature. In this note we announce that if M has (
n - 1) principal curvatures with the same sign everywhere, then M is isomet
ric to a Clifford Torus S-1 (root1/n) x Sn-1 (rootn-1/n).