Hypersurfaces of ${\mathbb{H}}^2\times {\mathbb{H}}^2$ with constant sectional curvature

Abstract

In this paper, we classify the hypersurfaces of ${\mathbb{H}}^2\times {\mathbb{H}}^2$ with constant sectional curvature. In contrast to ${\mathbb{S}}^2\times {\mathbb{S}}^2$, the resulting examples for ${\mathbb{H}}^2\times {\mathbb{H}}^2$ exhibit more diversity, and we construct a special example with non-constant product angle function. For ${\mathbb{S}}^2\times {\mathbb{S}}^2$, however, the product angle function of any constant sectional curvature hypersurface is identically zero. As a byproduct, we classify the hypersurfaces of ${\mathbb{H}}^2\times {\mathbb{H}}^2$ with constant product angle function and constant mean curvature (or constant scalar curvature).

46 pages. Comments are welcome. arXiv admin note: substantial text overlap with arXiv:2302.00466