SUBMANIFOLDS WITH PARALLEL MEAN-CURVATURE VECTOR IN SPHERES

Let M(n) be a compact submanifold of a sphere with parallel mean curva ture vector h. We introduce a tensor phi, related to h and the second fundamental form, and show that if phi satisfies a certain inequality, then phi=0 and M(n) is totally umbilic or equality holds. The estimat e is sharp in the sense that we describe all M(n) that satisfy this eq uality.