A \(3\)-manifold with no real projective structure (Q318718)

A real projective structure on a manifold is an atlas consisting of charts in real projective space and transition maps that are restrictions of projective transformations. Examples of real projective structures are the familiar spherical, Euclidean and hyperbolic structures of constant curvature.NEWLINENEWLINEEvery closed surface admits a constant curvature Riemannian metric and hence a real projective structure. It is natural to ask whether each closed 3-dimensional manifold also has a real projective structure. The connected sum of two copies of 3-dimensional real projective space has a geometric structure in the sense of Thurston. However, this note shows that it does not admit a real projective structure. It remains a tantalising open question whether there is a universal geometry for 3-manifolds.