# 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.

## Key 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 $M$.