Projective and conformal closed manifolds with a higher-rank lattice action

Abstract

We prove global results about actions of cocompact lattices in higher-rank simple Lie groups on closed manifolds endowed with either a projective class of connections or a conformal class of pseudo-Riemannian metrics of signature (p, q), with \(\min (p,q) \ge 2\). Building on a recent article [17], provided that such a structure is locally equivalent to its model \({\textbf{X}}\), the main question treated here is the completeness of the associated \((G,{\textbf{X}})\)-structure. Because of the similarities between the model spaces of projective geometry and non-Lorentzian conformal geometry, a number of arguments apply in both contexts. We therefore present the proofs in parallel. The conclusion is that in both cases, when the real rank is maximal, the manifold is globally equivalent to either the model space \({\textbf{X}}\) or its double cover.