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

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)>1. In the continuity of a recent article, provided that such a structure is locally equivalent to its model X, the main question treated here is the completeness of the associated (G,X)-structure. The similarities between the model spaces of non-Lorentzian conformal geometry and projective geometry make that lots of arguments are valid for both cases, and we expose 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 X or its double cover.