HYPERSURFACES IN SPACE FORMS SATISFYING THE CONDITION $L_k x = Ax + b$

Abstract

We study hypersurfaces either in the sphere $\mathbb{S}^{n+1}$ or in the hyperbolic space $\mathbb{H}^{n+1}$ whose position vector $x$ satisfies the condition $L_k x = Ax + b$, where $L_k$ is the linearized operator of the $(k+1)$-th mean curvature of the hypersurface for a fixed $k = 0, \ldots, n−1$, $A \in \mathbb{R}^{(n+2) \times (n+2)}$ is a constant matrix and $b \in \mathbb{R}^{n+2}$ is a constant vector. For every $k$, we prove that when $A$ is self-adjoint and $b=0$, the only hypersurfaces satisfying that condition are hypersurfaces with zero $(k+1)$-th mean curvature and constant $k$-th mean curvature, and open pieces of standard Riemannian products of the form $\mathbb{S}^m(\sqrt{1-r^2}) \times \mathbb{S}^{n-m}(r) \subset \mathbb{S}^{n+1}$, with $0 \lt r \lt 1$, and $\mathbb{H}^m(−\sqrt{1+r^2}) \times \mathbb{S}^{n-m}(r) \subset \mathbb{H}^{n+1}$, with $r \gt 0$. If $H_k$ is constant, we also obtain a classification result for the case where $b \neq 0$.