Convex projective manifolds, symmetric spaces and geometric decompositions

Abstract

We prove that if a closed, indecomposable, properly convex real projective $4$-manifold is geometric or admits a geometric decomposition in the sense of Thurston, then every piece is real hyperbolic. This extends a theorem of Benoist to dimension four. Moreover, we build orientable (non-hyperbolic) $4$-manifolds of the above type, with arbitrary positive, even, Euler characteristic. Along the way, we characterise the compact locally symmetric spaces that virtually support properly convex real projective structures.