Hypersurfaces with constant scalar or mean curvature in a unit sphere

Let M be an n(n >= 3)-dimensional complete connected hypersurface in a unit sphere S(n+1)(1). In this paper, we show that (1) if M has non-zero mean curvature and constant scalar curvature n(n-1)r and two distinct principal curvatures, one of which is simple, then M is isometric to the Riemannian product S(1)(root 1 - c(2)) x S(n-1)(c), c(2) = n-2/nr if r >= n-2/n-1 and S <= (n-1) n(r-1)+2/n-2 + n-2/n(r-1)+2. (2) if M has non-zero constant mean curvature and two distinct principal curvatures, one of which is simple, then M is isometric to the Riemannian product S(1)(root 1 - c(2)) x S(n-1)(c), c(2) = n-2/nr if one of the following conditions is satisfied: (i) r >= n-2/n-1 and S <= (n-1) n(r-1)+2/n-2 + n-2/n(r-1)+2; or (ii) r > 1-2/n, r not equal n-2/n-1 and S <= (n-1) n(r-1)+2/n-2 + n-2/n(r-1)+2, where S is the squared norm of the second fundamental form of M.