Closed ray affine manifolds

TL;DR

This paper studies closed affine manifolds with a specific rank one ray structure, showing they are either complete or have a developing map covering the complement of an affine subspace, and explores conditions for completeness and automorphism group actions.

Contribution

It extends Fried's geometric picture to rank one ray structures and proves that parallel volume implies completeness, also analyzing automorphism group actions.

Findings

Closed manifolds are either complete or their developing map covers the complement of an affine subspace.

Parallel volume condition implies the manifold is necessarily complete.

Automorphism group acts non properly when the manifold is complete.

Abstract

We consider closed manifolds that possess a so called rank one ray structure. That is a (flat) affine structure such that the linear part is given by the products of a diagonal transformation and a commuting rotation. We show that closed manifolds with a rank one ray structure are either complete or their developing map is a cover onto the complement of an affine subspace. This result extends the geometric picture given by Fried on closed similarity manifolds. We prove, in the line of Markus conjecture, that if the rank one ray geometry has parallel volume, then closed manifolds are necessarily complete. Finally, we show that the automorphism group of a closed manifold acts non properly when the manifold is complete.