# Spaces of locally homogeneous affine surfaces

## TL;DR

This paper studies the topological structure of spaces of locally homogeneous affine surfaces, classifying them based on Ricci tensor properties and group actions, extending previous classification results.

## Contribution

It determines the topology of spaces of Type A and B affine models, relating them to Ricci tensor rank and flatness, building on Opozda's classification.

## Key findings

- Topology of Type A models depends on Ricci tensor rank
- Type B models are either flat or have alternating Ricci tensor
- Provides a detailed topological classification of affine surface spaces

## Abstract

We examine the topology of various spaces of locally homogeneous affine manifolds which arise from the classification result of Opozda [B. Opozda, A classification of locally homogeneous connections on 2-dimensional manifolds, Differential Geom. Appl. 21 (2004), 173-198.] as orbits of the action of $GL(2,\mathbb{R})$ (Type $\mathcal{A}$) and the $ax+b$ group (Type $\mathcal{B}$). We determine the topology of the spaces of Type $\mathcal{A}$ models in relation to the rank of the Ricci tensor. We determine the topology of the spaces of Type $\mathcal{B}$ models which either are flat or where the Ricci tensor is alternating.