Semisimple Weakly Symmetric Pseudo--Riemannian Manifolds

We develop the classification of weakly symmetric pseudo–riemannian manifolds $G\mathrm{/}H$ where $G$ is a semisimple Lie group and $H$ is a reductive subgroup. We derive the classification from the cases where $G$ is compact, and then we discuss the (isotropy) representation of $H$ on the tangent space of $G\mathrm{/}H$ and the signature of the invariant pseudo–riemannian metric. As a consequence we obtain the classification of semisimple weakly symmetric manifolds of Lorentz signature $(n-1,1)$ and trans–Lorentz (conformal Lorentz) signature $(n-2,2)$.