The half-space model of pseudo-hyperbolic space

In this thesis we develop a half-space model $\mathcal{H}^{p,q}$ for the pseudo-hyperbolic space $\mathbb{H}^{p,q}$ for any $p,q$ with $p\ge1$. 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. Finally, we describe the action of the isometries of $\mathbb{H}^{p,q}$ on $\mathcal{H}^{p,q}$.