On hypersurfaces with two distinct principal curvatures in a unit sphere

Abstract

We investigate the immersed hypersurfaces in a unit sphere S n + 1 ( 1 ) . By using Otsuki's idea, we obtain the local and global classification results for immersed hypersurfaces in S n + 1 ( 1 ) of constant m-th mean curvature and two distinct principal curvatures of multiplicities n − 1 , 1 (in the local version, we assume that the principal curvatures are non-zero when m ⩾ 2 ). As the result, we prove that any local hypersurface in S n + 1 ( 1 ) of constant mean curvature and two distinct principal curvatures is an open part of a complete hypersurface of the same curvature properties. The corresponding result does not hold for m-th mean curvature when m ⩾ 2 .