Hypersurfaces with constant mean curvature in a real space form

Let Mn be an n (n ≥ 3)-dimensional complete connected and oriented hypersurface in M n+1(c)(c ≥0) with constant mean curvature H and with two distinct principal curvatures, one of which is simple. We show that (1) if c = 1 and the squared norm of the second fundamental form of Mn satisfies a rigidity condition (1.3), then Mn is isometric to the Riemannian product S1( √1-a2) × Sn−1(a); (2) if c = 0, H≠ 0 and the squared norm of the second fundamental form of Mn satisfies S ≥ n2H2/(n −1), then Mn is isometric to the Riemannian product S n−1 (a) ×R or S1(a) × Rn−1.