Weakly symmetric pseudo–Riemannian nilmanifolds

Abstract

In an earlier paper we developed the classification of weakly symmetric pseudo–Riemannian manifolds $G/H$, where $G$ is a semisimple Lie group and $H$ is a reductive subgroup. We derived the classification from the cases where $G$ is compact. As a consequence we obtained the classification of semisimple weakly symmetric manifolds of Lorentz signature $(n-1,1)$ and trans-Lorentzian signature $(n-2,2)$. Here we work out the classification of weakly symmetric pseudo-Riemannian nilmanifolds $G/H$ from the classification for the case $G=N \rtimes H$ with $H$ compact and $N$ nilpotent. It turns out that there is a plethora of new examples that merit further study. Starting with that Riemannian case, we see just when a given involutive automorphism of $H$ extends to an involutive automorphism of $G$, and we show that any two such extensions result in isometric pseudo-Riemannian nilmanifolds. The results are tabulated in the last two sections of the paper.