SUBMANIFOLDS OF CONSTANT SCALAR CURVATURE IN A HYPERBOLIC SPACE FORM

Abstract

Let $M^n$ be a closed submanifold immersed into a real hyperbolic space form $\Bbb H^{n+p}$ of constant curvature $-1$. Denote by $R$ the normalized scalar curvature of $M^n$ and by $H$ the mean curvature of $M^n$. Suppose that $R$ is constant and bigger than or equal to $-1$. We first extend Cheng-Yau's technique to higher codimensional cases. Then, for $M^n$ with parallel normalized mean curvature vector field, we show that, if $H$ satisfies a certain inequality, then $M^n$ is totally umbilical or the equality part holds. We describe all $M^n$ whose $H$ satisfies this equality.