Affine Killing complete and geodesically complete homogeneous affine surfaces (Q2414814)

Let $M$ be a connected smooth manifold equipped with a torsion-free connection $\nabla$ on the tangent bundle of $M$. Then the pair $(M,\nabla)$ is said to be an affine manifold. An affine manifold is said to be geodesically complete, if all affine geodesics extend to all time. It is said to be affine Killing complete, if the integral curves for any affine Killing vector field extend to all time. \par The aim of the present paper is to examine geodesic completeness (resp. Killing completeness). The authors emphasize that geodesic completeness (resp. affine Killing completeness) is equivalent to prolonging of some non-linear ODEs, which can be quite unmanageable even in the homogeneous setting. Consequently, instead of a direct approach, they use the solution space of the affine quasi-Einstein equation.