Compact hypersurfaces in a unit sphere

We study curvature structures of compact hypersurfaces in the unit sphere Sn+1 (1) with two distinct principal curvatures. First of all, we prove that the Riemannian product S-1(root 1-c(2)) x Sn-1(c) is the only compact hypersurface in Sn+1 (1) with two distinct principal curvatures, one of which is simple and satisfies 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 n(n - 1)r is the scalar curvature of hypersurfaces and c(2) = (n - 2)/nr. This generalized the result of Cheng, where the scalar curvature is constant is assumed. Secondly, we prove that the Riemannian product S-1(root 1-c(2)) x Sn-1 (c) is the only compact hypersurface with non-zero mean curvature in Sn+1 (1) with two distinct principal curvatures, one of which is simple and satisfies r > 1 - 2/n and S <= (n - 1) n(r - 1) + 2/n - 2 + n - 2/n(r - 1) + 2. This gives a partial answer for the problem proposed by Cheng.