# Geodesic completeness and the quasi-Einstein equation for locally homogeneous affine surfaces

## TL;DR

This paper investigates the geodesic completeness of Type A affine surfaces by analyzing their projective flatness and applying the quasi-Einstein equation, revealing new insights into their geometric structure.

## Contribution

It establishes that Type A affine surfaces are linearly strongly projectively flat and links this property to their geodesic completeness using the quasi-Einstein equation.

## Key findings

- Type A affine surfaces are linearly strongly projectively flat
- The quasi-Einstein equation helps determine geodesic completeness
- New criteria for geodesic completeness in affine surfaces

## Abstract

Let $\mathcal{M}$ be a Type $\mathcal{A}$ affine surface. We show that $\mathcal{M}$ is linearly strongly projectively flat. We use the quasi-Einstein equation together with the condition that $\mathcal{M}$ is strongly projectively flat to examine to examine the geodesic completeness of $\mathcal{M}$.