Isotropy of semisimple group actions on manifolds with geometric structure

We prove results concerning isotropy of noncompact semisimple Lie group actions that preserve pseudo-Riemannian or affine structures on compact or finite volume manifolds. For example, if a Lie group G acts locally faithfully on a connected compact complete affine manifold M and preserves the affine structure, then the connected component of the identity of each isotropy subgroup is a compact extension of a solvable group. This is a consequence of the result that nontrivial affine actions of connected groups locally isomorphic to SL₂[inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] on such manifolds cannot have fixed points. We also prove that if G is connected semisimple without compact factors and acts isometrically and topologically transitively on a connected finite volume pseudo-Riemannian M, then the action must be (everywhere) locally free. This answers affirmatively a special case of a conjecture by G. D'Ambra and M. Gromov. With only the assumption that no factor of G acts trivially, local freeness is also assured when M is compact and complete.