Characterization of totally umbilical hypersurfaces

Abstract

This paper gives a sufficient condition for a complete hypersurface of a Riemannian manifold of constant curvature to be umbilical. The condition will be given by an inequality which is established between the length of the second fundamental tensor and the mean curvature. K. Nomizu and B. Smyth in [3] established a formula for the Laplacian of the second fundamental form of a hypersurface M immersed with constant mean curvature in a space M of constant sectional curvature c. Later, M. Okumura in [4] characterized under certain conditions a totally umbilical hypersurface of a Riemannian manifold of nonnegative constant curvature by an inequality between the length of the second fundamental tensor and the mean curvature of the hypersurface. In the present article we prove the following theorem. THEOREM A. Let M be an n-dimensional (n > 3) connected complete hypersurface immersed with constant mean curvature in an (n + 1)-dimensional Riemannian manifold M of positive constant curvature c. If the second fundamental tensor L satisfies trace L2 cn or M is totally geodesic. 1. Preliminaries. Let M be an (n + 1)-dimensional Riemannian manifold of constant curvature c. Let q: M -, M be an isometric immersion of an n-dimensional manifold M into M. In what follows we identify M with p(M) and p E M with (p(p) E rp(M) c M. The tangent space TpM is also identified with a subspace of T9,(p)M. The Riemannian metric g of M is induced from the Riemannian metric Received by the editors December 27, 1979 and, in revised form, February 26, 1980. AMS (MOS) subject classifications (1970). Primary 53C40, 53C20.