The half-space model of pseudo-hyperbolic space

TL;DR

This paper introduces a half-space model for pseudo-hyperbolic space that facilitates understanding its geometric structures and symmetries, extending classical models to a broader pseudo-Riemannian context.

Contribution

It develops a new half-space model for $ ext{H}^{p,q}$, describing its geometric features and isometry group, generalizing the classical hyperbolic space models.

Findings

Provides an isometric embedding of the half-space model

Describes geodesics and totally geodesic submanifolds

Explains boundary at infinity interpretation

Abstract

In this note we develop a half-space model for the pseudo-hyperbolic space $\mathbb{H}^{p,q}$, for any $p,q$ with $p\geq 1$. 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.