The geometry of locally symmetric affine surfaces

TL;DR

This paper classifies and analyzes the local geometry of affine surfaces that are locally symmetric, identifying six distinct types and exploring their geodesic structures, including Lorentzian analogues.

Contribution

It provides a classification of six non-isomorphic local geometries of symmetric affine surfaces and examines their geodesic structures, including Lorentzian cases.

Findings

Six non-isomorphic local geometries identified

Realization of geometries as Type A, B, C using Opozda's result

Analysis of geodesic structures and Lorentzian analogues

Abstract

We examine the local geometry of affine surfaces which are locally symmetric. There are 6 non-isomorphic local geometries. We realize these examples as Type A, Type B, and Type C geometries using a result of Opozda and classify the relevant geometries up to linear isomorphism. We examine the geodesic structures in this context. Particular attention is paid to the Lorentzian analogue of the hyperbolic plane and to the pseudosphere.