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The half-space model of pseudo-hyperbolic space

Andrea Seppi, Enrico Trebeschi

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.

The half-space model of pseudo-hyperbolic space

Abstract

In this note we develop a half-space model for the pseudo-hyperbolic space , for any with . This half-space model embeds isometrically onto the complement of a degenerate totally geodesic hyperplane in . 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.

Paper Structure

This paper contains 24 sections, 21 theorems, 59 equations, 6 figures.

Table of Contents

  1. Introduction
  2. First definitions and properties
  3. The half-space model
  4. An isometric embedding
  5. Symmetries
  6. Totally geodesic submanifolds
  7. The geodesic equations
  8. Totally geodesic hypersurfaces
  9. The general classification
  10. Geodesics
  11. Lightlike geodesics
  12. A preliminary computation
  13. Timelike geodesics
  14. Spacelike geodesics
  15. The boundary at infinity
  16. ...and 9 more sections

Key Result

Proposition 2.5

There exists an isometric embedding If $q=0$, $\iota_{p,q}$ is surjective. Otherwise, its image is the complement of a totally geodesic degenerate hyperplane in $\mathbb{H}^{p,q}$.

Figures (6)

  • Figure 1: The totally geodesic quadric hypersurfaces in $\mathcal{H}^{2,1}$ (left) and $\mathcal{H}^{1,2}$ (right).
  • Figure 2: The lightcone from a point in $\mathcal{H}^{2,1}$ (left) and $\mathcal{H}^{1,2}$ (right).
  • Figure 3: A timelike geodesic (in red) and the four types of spacelike geodesics (blue).
  • Figure 4: The compactification of $\partial\mathcal{H}^{2,1}$, which is a copy of $\mathbb{R}^{1,1}$ represented by the interior of the diamond, inside $\partial_\infty\mathcal{H}^{2,1}$. The lines at $\pm 45^\circ$ represent the degenerate affine subspaces in $\mathbb{R}^{1,1}$, and each of them is compactified to a different point. The point $\infty$ then corresponds to the vertices of the diamond. The identifications of the sides clearly give the topology of a torus on $\partial_\infty\mathcal{H}^{2,1}$.
  • Figure 5: Horizontal horospheres, and wedges of hyperplanes.
  • ...and 1 more figures

Theorems & Definitions (59)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Proposition 2.5
  • proof
  • Lemma 2.6
  • proof
  • Proposition 3.1
  • proof
  • ...and 49 more