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We prove global results about actions of cocompact lattices in higher-ranksimple Lie groups on closed manifolds endowed with either a projective class ofconnections or a conformal class of pseudo-Riemannian metrics of signature$(p,q)$, with $\min(p,q) \geq 2$. In the continuity of a recent article,provided that such a structure is locally equivalent to its model $\mathbf{X}$,the main question treated here is the completeness of the associated$(G,\mathbf{X})$-structure. Because of the similarities between the modelspaces of projective geometry and non-Lorentzian conformal geometry, a numberof arguments apply in both contexts. We therefore present the proofs inparallel. The conclusion is that in both cases, when the real-rank is maximal,the manifold is globally equivalent to either the model space $\mathbf{X}$ orits double cover.

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