Geodesic completeness for Type~A surfaces

TL;DR

This paper investigates the geodesic completeness of Type A affine surfaces, providing criteria to determine when such surfaces modeled by constant Christoffel symbols are geodesically complete or incomplete.

Contribution

It offers a classification and a method to decide if a set of constant Christoffel symbols can model a geodesically complete surface.

Findings

Some Type A models are geodesically complete.

Some models admit incomplete geodesics but can model complete surfaces.

Certain models do not correspond to any complete surface.

Abstract

Type A surfaces are the locally homogeneous affine surfaces which can be locally described by constant Christoffel symbols. We address the issue of the geodesic completeness of these surfaces: we show that some models for Type A surfaces are geodesically complete, that some others admit an incomplete geodesic but model geodesically complete surfaces, and that there are also others which do not model any complete surface. Our main result provides a way of determining whether a given set of constant Christoffel symbols can model a complete surface.