Pinching theorems of hypersurfaces in a unit sphere

Let MnMn be a complete hypersurface in Sn+1(1)Sn+1(1) with constant mean curvature. Assume that MnMn has n−1n−1 principal curvatures with the same sign everywhere. We prove that if RicM≤C−(H)RicM≤C−(H), either S⩽S+(H)S⩽S+(H) or RicM⩾0RicM⩾0 or the fundamental group of MnMn is infinite, then S   is constant, S=S+(H)S=S+(H) and MnMn is isometric to a Clifford torus S1(1−r2)×Sn−1(r) with r2⩾n−1n. These rigidity theorems are still valid for compact hypersurface without constancy condition on the mean curvature.