The half-space model of pseudo-hyperbolic space

Abstract

In this note we develop a half-space model for the pseudo-hyperbolic space ${\mathbb{H}}^{p,q}$, for any $p,q$ with $p\ge 1$. This half-space model embeds isometrically onto the complement of a degenerate totally geodesic hyperplane in ${\mathbb{H}}^{p,q}$. We describe the geodesics, the totally geodesic submanifolds, the horospheres, the isometry group in the half-space model, and we explain how to interpret the boundary at infinity in this setting.

24 pages, 6 figures. Added Lemma 6.2 and additional details in the proofs of Propositions 2.5 and 5.1; other minor improvements and corrections