# Homogeneous surfaces admitting invariant connections

## TL;DR

This paper classifies simply connected homogeneous surfaces that admit invariant affine connections, identifying six surfaces with multiple invariant connections and four with exactly one, enriching understanding of geometric structures on surfaces.

## Contribution

It provides a complete classification of simply connected homogeneous surfaces with invariant affine connections, detailing the number of such connections for each surface.

## Key findings

- Six surfaces admit more than one invariant connection.
- Four surfaces admit exactly one invariant connection.
- Complete classification of invariant affine connections on these surfaces.

## Abstract

We compute all the simply connected homogeneous and infinitesimally homogeneous surfaces admitting one or more invariant affine connections. We find exactly six non equivalent simply connected homogeneous surfaces admitting more than one invariant connections and four classes of simply connected homogeneous surfaces admitting exactly one invariant connection.