Quasihomogeneous analytic affine connections on surfaces

TL;DR

This paper classifies certain real-analytic affine connections on surfaces that are locally homogeneous in parts but not globally, proving their existence and providing a detailed local classification.

Contribution

It provides the first classification of quasihomogeneous torsion-free real-analytic affine connections on surfaces, demonstrating their existence and characterizing their local structure.

Findings

Existence of quasihomogeneous affine connections on surfaces.

Classification of germs of such connections near a point.

Identification of local models for these connections.

Abstract

We classify torsion-free real-analytic affine connections on compact oriented real-analytic surfaces which are locally homogeneous on a nontrivial open set, without being locally homogeneous on all of the surface. In particular, we prove that such connections exist. This classification relies in a local result that classifies germs of torsion-free real-analytic affine connections on a neighborhood of the origin in the plane which are quasihomogeneous, in the sense that they are locally homogeneous on an open set containing the origin in its closure, but not locally homogeneous in the neighborhood of the origin.