Submanifolds with constant scalar curvature in a space form

Abstract

We deal with complete submanifolds Mn having constant positive scalar curvature and immersed with parallel normalized mean curvature vector field in a Riemannian space form Qcn+p of constant sectional curvature c∈{1,0,−1}. In this setting, we show that such a submanifold Mn must be either totally umbilical or isometric to a Clifford torus S1(1−r2)×Sn−1(r), when c=1, a circular cylinder R×Sn−1(r), when c=0, or a hyperbolic cylinder H1(−1+r2)×Sn−1(r), when c=−1. This characterization theorem corresponds to a natural improvement of previous ones due to Alías, García-Martínez and Rigoli [2], Cheng [4] and Guo and Li [6].