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 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 $M$.