Symmetric affine surfaces with torsion

TL;DR

This paper classifies symmetric affine surfaces with torsion, extending previous work to include non-zero torsion cases, and explores their local homogeneity and geometric structures.

Contribution

It provides a complete classification of symmetric affine surfaces with torsion, generalizing prior results to include non-zero torsion and local homogeneity cases.

Findings

Classified symmetric affine surfaces with parallel torsion

Extended classification to non-parallel torsion under local homogeneity

Found all such geometries are locally homogeneous

Abstract

We study symmetric affine surfaces which have non-vanishing torsion tensor. We give a complete classification of the local geometries possible if the torsion is assumed parallel. This generalizes a previous result of Opozda in the torsion free setting; these geometries are all locally homogeneous. If the torsion is not parallel, we assume the underlying surface is locally homogeneous and provide a complete classification in this setting as well.