Classification of hypersurfaces with constant principal curvatures in ${\mathbb{H}}^m\times {\mathbb{H}}^n$

Abstract

In this paper, we study the hypersurfaces in ${\mathbb{H}}^m\times {\mathbb{H}}^n$ ($m\ge 3,n\ge 2$) with constant principal curvatures. Let $g$ be the number of distinct constant principal curvatures. First, we classify all such hypersurfaces with $g\le 2$. Then, we prove that a hypersurface with constant principal curvatures and constant product angle function has $g\le 3$, and we obtain a complete classification of these hypersurfaces. As a corollary, we classify the isoparametric hypersurfaces in ${\mathbb{H}}^m\times {\mathbb{H}}^n$ ($m\ge 3,n\ge 2$) with constant principal curvatures.

arXiv comment39 pages. Comments are welcome