Closed Affine Manifolds with an Invariant Line

TL;DR

This paper investigates the structure of closed affine manifolds with invariant lines, proving that certain holonomy conditions prevent the developing image from intersecting the invariant line, thus addressing their non-existence.

Contribution

It introduces new conditions under which closed affine manifolds with invariant lines cannot exist, extending previous understanding of affine holonomy and developing map properties.

Findings

If the affine holonomy acts purely by translations on the invariant line, the developing image cannot meet this line.

Constructs large open subsets where the developing map is a diffeomorphism onto its image.

Provides a modified proof that radiant manifolds cannot have their fixed point in the developing image.

Abstract

A closed affine manifold is a closed manifold with coordinate patches into affine space whose transition maps are restrictions of affine automorphisms. Such a structure gives rise to a local diffeomorphism from the universal cover of the manifold to affine space that is equivariant with respect to a homomorphism from the fundamental group to the group of affine automorphisms. The local diffeomorphism and homomorphism are referred to as the developing map and holonomy respectively. In the case where the linear holonomy preserves a common vector, certain `large' open subsets upon which the developing map is a diffeomorphism onto its image are constructed. A modified proof of the fact that a radiant manifold cannot have its fixed point in the developing image is presented. Combining these results, this paper addresses the non-existence of certain closed affine manifolds whose holonomy leaves invariant an affine line. Specifically, if the affine holonomy acts purely by translations on the invariant line, then the developing image cannot meet this line.