  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.
