A classification of locally homogeneous affine connections on compact surfaces

TL;DR

This paper classifies locally homogeneous affine connections on compact surfaces, showing they are either flat, Riemannian with constant curvature, or quotients of translation-invariant connections, advancing understanding of geometric structures on surfaces.

Contribution

It provides a complete classification of locally homogeneous affine connections on compact surfaces, identifying three distinct types and their geometric properties.

Findings

Connections are either torsion-free and flat

Connections are Levi-Civita of constant curvature metrics

Connections are quotients of translation-invariant connections

Abstract

We classify the affine connections on compact orientable surfaces for which the pseudogroup of local isometries acts transitively. We prove that such a connection is either torsion-free and flat, the Levi-Civita connection of a Riemannian metric of constant curvature or the quotient of a translation-invariant connection in the plane.