Homogeneous affine surfaces: Moduli spaces

TL;DR

This paper studies the classification and structure of moduli spaces of non-flat homogeneous affine surfaces, providing explicit invariants for Type A and describing the topology of Type B moduli spaces.

Contribution

It introduces complete invariants for Type A surfaces and characterizes the moduli space topology for Type B surfaces, advancing the understanding of affine surface classifications.

Findings

Complete invariants for Type A surfaces based on Ricci tensor rank

Moduli space of Type B surfaces is a simply connected 4-dimensional manifold

Second Betti number of the Type B moduli space is 1

Abstract

We analyze the moduli space of non-flat homogeneous affine connections on surfaces. For Type $\mathcal{A}$ surfaces, we write down complete sets of invariants that determine the local isomorphism type depending on the rank of the Ricci tensor and examine the structure of the associated moduli space. For Type $\mathcal{B}$ surfaces which are not Type $\mathcal{A}$ we show the corresponding moduli space is a simply connected real analytic 4-dimensional manifold with second Betti number equal to $1$.