# IJPAM: Volume 72, No. 3 (2011)

ON SUBMANIFOLDS WITH CONSTANT MEAN CURVATURE
IN A REAL SPACE FORM

Shingo Kikuchi, Yoshio Matsuyama
Department of Mathematics
Chuo University
1-13-27, Kasuga, Bunkyo-ku, Tokyo, 112-8551, JAPAN

Abstract. The purpose of this paper is to classify submanifolds with constant mean curvature in a real space form. We put the squared norm of the second fundamental form and . Denote by and the squares of the positive roots of the equations

respectively. We prove the following: First, le be a complete, connected and orientable submanifold with nonzero constant mean curvature in . If satisfies for all , then lies in a totally geodesic submanifold of , and and is totally umbilic. Next, let be a complete, connected and orientable hypersurface with constant mean curvature in . Assume that for all . Then (i) either and is totally umbilic or . (ii) if and only if is isometric to for some . Moreover, we prove a gereralization of this result of the hypersurface in a hyperbolic space.

AMS Subject Classification: 53C40, 53B25

Key Words and Phrases: reduction of the codimension, parallel second fundamental form, totally umbilic