Compact Lorentz manifolds with local symmetry

Abstract

We prove a structure theorem for compact aspherical Lorentz manifolds with abundant local symmetry. If M is a compact, aspherical, real-analytic, complete Lorentz manifold such that the isometry group of the universal cover has semisimple identity component, then the local isometry orbits in M are roughly fibers of a fiber bundle. A corollary is that if M has a dense local isometry orbit then M is locally homogeneous. The main result is analogous to a theorem of Farb and Weinberger on compact aspherical Riemannian manifolds, and an exposition of their arguments on rational cohomological dimension is included. Some aspects of dynamics on Lorentz manifolds are also presented, including totally geodesic, lightlike, codimension-one foliations associated to unbounded sequences of isometries.