Closed Affine Manifolds with an Invariant Line

Abstract

Radiant manifolds are affine manifolds whose holonomy preserves a point. Here we discuss certain properties of closed affine manifolds whose holonomy preserves an affine line. Particular attention is given to the case wherein the holonomy acts on the invariant line by translations and reflections. We show that in this case, the developing image must avoid the translation invariant line providing a generalization to the well known fact that closed radiant affine manifolds cannot have their fixed points inside the developing image. We conclude by generalizing this result to translation and reflection invariant proper subspaces of the holonomy.