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Closed similarity Lorentzian affine manifolds

Author: Tsemo Aristide
Journal: Proc. Amer. Math. Soc. 132 (2004), 3697-3702
MSC (2000): Primary 53C30, 53C50
Published electronically: July 22, 2004
MathSciNet review: 2084093
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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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Additional Information

Tsemo Aristide
Affiliation: The International Center for Theoretical Physics, Strada Costiera, 11, Trieste, Italy
Address at time of publication: 3738, Avenue de Laval, Appt. 106, Montreal, Canada H2X 3C9

Received by editor(s): April 28, 2001
Published electronically: July 22, 2004
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2004 American Mathematical Society