Locally homogeneous rigid geometric structures on surfaces

TL;DR

This paper investigates locally homogeneous rigid geometric structures on surfaces, proving flatness of projective connections on compact surfaces and completeness and homogeneity of unimodular affine connections on tori.

Contribution

It establishes that locally homogeneous projective connections on compact surfaces are flat and that unimodular affine connections on tori are complete and, after finite covers, homogeneous.

Findings

Locally homogeneous projective connections on compact surfaces are flat.

Unimodular affine connections on tori are complete.

Such connections are globally homogeneous after finite covers.

Abstract

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 $M$.