Closed similarity Lorentzian affine manifolds

A $Sim(n-1,1)$ affine manifold is an $n$-dimensional affine manifold whose linear holonomy lies in the similarity Lorentzian group but not in the Lorentzian group. In this paper, we show that a compact $Sim(n-1,1)$ affine manifold is incomplete. Let $\langle,\rangle_L$be the Lorentz form, and $q$ the map on ${\mathbb R}^n$ defined by $q(x)=\langle x,x\rangle_L$. We show that for a compact radiant $Sim(n-1,1)$affine manifold $M$, if a connected component $C$ of ${\mathbb R}^n-q^{-1}(0)$ intersects the image of the universal cover of $M$ by the developing map, then either $C$ or a connected component of $C-H$, where $H$ is a hyperplane, is contained in this image.