Locally homogeneous rigid geometric structures on surfaces

We study locally homogeneous rigid geometric structures on surfaces. We show that a locally homogeneous projective connection on a compact surface is flat. We also show that a locally homogeneous unimodular affine connection ${nabla}$ on a two dimensional torus is complete and, up to a finite cover, homogeneous. Let ${nabla}$ be a unimodular real analytic affine connection on a real analytic compact connected surface M. If ${nabla}$ is locally homogeneous on a nontrivial open set in M, we prove that ${nabla}$ is locally homogeneous on all of M.