Locally Isotropic Pseudo-Riemannian Manifolds

Locally isotropic pseudo-Riemannian manifolds are known to be locally symmetric; this result is due to Wolf. In the Riemannian setting one proof, due to Szab\'o, uses spectral properties of the so-called Szab\'o operator. In this paper we extend Szab\'o's method to the pseudo-Riemannian setting, obtaining results comparable to those of Wolf. Most results concerning the Szab\'o operator in the pseudo-Riemannian setting are presented using methods of algebraic topology. This paper exploits the polynomial nature of the Szab\'o operator along with its behavior over the nullcone, both of which have received little attention this far.