Closed similarity Lorentzian affine manifolds

Aristide, Tsemo

doi:10.1090/S0002-9939-04-07560-4

Closed similarity Lorentzian affine manifolds
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Proc. Amer. Math. Soc. 132 (2004), 3697-3702 Request permission

Abstract:

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.

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