We consider closed manifolds that possess a so called rank one ray structure. That is a (flat) affine structure such that the linear part of the holonomy is given by 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. 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 if the automorphism group of a closed manifold is non-compact then the manifold is complete.