Closed similarity Lorentzian affine manifolds
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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.References
- Yves Carrière, Autour de la conjecture de L. Markus sur les variétés affines, Invent. Math. 95 (1989), no. 3, 615–628 (French, with English summary). MR 979369, DOI 10.1007/BF01393894
- David Fried, Closed similarity manifolds, Comment. Math. Helv. 55 (1980), no. 4, 576–582. MR 604714, DOI 10.1007/BF02566707
- David Fried, William Goldman, and Morris W. Hirsch, Affine manifolds with nilpotent holonomy, Comment. Math. Helv. 56 (1981), no. 4, 487–523. MR 656210, DOI 10.1007/BF02566225
- Claude Godbillon, Feuilletages, Progress in Mathematics, vol. 98, Birkhäuser Verlag, Basel, 1991 (French). Études géométriques. [Geometric studies]; With a preface by G. Reeb. MR 1120547
- William M. Goldman, Projective structures with Fuchsian holonomy, J. Differential Geom. 25 (1987), no. 3, 297–326. MR 882826
- Jean-Louis Koszul, Variétés localement plates et convexité, Osaka Math. J. 2 (1965), 285–290 (French). MR 196662
- Tsemo, A. Thèse Université de Montpellier II, (1999).
- Joseph A. Wolf, Spaces of constant curvature, McGraw-Hill Book Co., New York-London-Sydney, 1967. MR 0217740
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
- Email: tsemoaristide@hotmail.com
- Received by editor(s): April 28, 2001
- Published electronically: July 22, 2004
- Communicated by: Ronald A. Fintushel
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 3697-3702
- MSC (2000): Primary 53C30, 53C50
- DOI: https://doi.org/10.1090/S0002-9939-04-07560-4
- MathSciNet review: 2084093