On hypersurfaces with two distinct principal curvatures in a unit sphere

We investigate the immersed hypersurfaces in a unit sphere Sn+1(1)Sn+1(1). By using Otsuki's idea, we obtain the local and global classification results for immersed hypersurfaces in Sn+1(1)Sn+1(1) of constant m  -th mean curvature and two distinct principal curvatures of multiplicities n−1,1n−1,1 (in the local version, we assume that the principal curvatures are non-zero when m⩾2m⩾2). As the result, we prove that any local hypersurface in Sn+1(1)Sn+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⩾2m⩾2.