Boundary Rigidity, and Non-Rigidity, of Projective Structures

We investigate the property of boundary rigidity for the projective structures associated to torsion-free affine connections on connected smooth manifolds with boundary. We show that these structures are generically boundary rigid, meaning that any automorphism of a generic projective structure that restricts to the identity on the boundary must itself be the identity. However, and in contrast with what happens for example for conformal structures, we show that there exist projective structures which are not boundary rigid. We characterise these non-rigid structures by the vanishing of a certain local projective invariant of the boundary.