  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 $M$.
