Complete hypersurfaces with [formula omitted] in a unit sphere

Abstract

By investigating hypersurfaces M n in the unit sphere S n + 1 ( 1 ) with H k = 0 and with two distinct principal curvatures, we give a characterization of torus S 1 ( k / n ) × S n − 1 ( ( n − k ) / n ) . We extend recent results of T. Hasanis et al. [T. Hasanis, A. Savas-Halilaj, T. Vlachos, Complete Minimal hypersurfaces in a sphere, Monatsh. Math. 145 (2005) 301–305] and T. Otsuki [T. Otsuki, Minimal hypersurfaces in a Riemannian manifold of constant curvature, Amer. J. Math. 92 (1970) 145–173].