A Ricci inequality for hypersurfaces in the sphere

Abstract

Let M n be a complete Riemannian manifold immersed isometrically in the unity Euclidean sphere $$\mathbb{S}^{n + 1} .$$ In [9], B. Smyth proved that if M n , n ≧ 3, has sectional curvature K and Ricci curvature Ric, with inf K > −∞, then sup Ric ≧ (n − 2) unless the universal covering $$\tilde M^n $$ of M n is homeomorphic to Rn or homeomorphic to an odd-dimensional sphere. In this paper, we improve the result of Smyth. Moreover, we obtain the classification of complete hypersurfaces of $$\mathbb{S}^{n + 1} .$$ with nonnegative sectional curvature.