There is a well developed theory of weakly symmetric Riemannian manifolds.
Here it is shown that several results in the Riemannian case are also valid for
weakly symmetric pseudo-Riemannian manifolds, but some require additional
hypotheses. The topics discussed are homogeneity, geodesic completeness, the
geodesic orbit property, weak symmetries, and the structure of the nilradical
of the isometry group.

In earlier papers we studied direct limits $(G,K) = \varinjlim (G_n,K_n)$ of
two types of Gelfand pairs. The first type was that in which the $G_n/K_n$ are
compact Riemannian symmetric spaces. The second type was that in which $G_n =
N_n\rtimes K_n$ with $N_n$ nilpotent, in other words pairs $(G_n,K_n)$ for
which $G_n/K_n$ is a commutative nilmanifold.