Linear Weingarten submanifolds immersed in a space form

Abstract

In this paper, we deal with complete linear Weingarten submanifolds $M^n$ immersed with parallel normalized mean curvature vector field in a Riemannian space form $\mathbf{Q}_c^{n+p}$ of constant sectional curvature $c$. Under an appropriated restriction on the norm of the traceless part of the second fundamental form, we show that such a submanifold $M^n$ must be either totally umbilical or isometric to a Clifford torus, if $c = 1$, a circular cylinder, if $c = 0$, or a hyperbolic cylinder, if $c = −1$. We point out that our results are natural generalizations of those ones obtained in [2] and [6].