Locally homogeneous rigid geometric structures on surfaces

International audienceWe 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 ∇ be a unimodular real analytic affine connection on a real an- alytic compact connected surface M. If ∇ is locally homogeneous on a nontrivial open set in M, we prove that ∇ is locally homogeneous on all of M