On the nonexistence of closed timelike geodesics in flat Lorentz 2-step nilmanifolds

Guediri, Mohammed

doi:10.1090/S0002-9947-02-03114-8

On the nonexistence of closed timelike geodesics in flat Lorentz 2-step nilmanifolds
HTML articles powered by AMS MathViewer

by Mohammed Guediri PDF

Trans. Amer. Math. Soc. 355 (2003), 775-786 Request permission

Abstract:

The main purpose of this paper is to prove that there are no closed timelike geodesics in a (compact or noncompact) flat Lorentz 2-step nilmanifold $N/\Gamma ,$ where $N$ is a simply connected 2-step nilpotent Lie group with a flat left-invariant Lorentz metric, and $\Gamma$ a discrete subgroup of $N$ acting on $N$ by left translations. For this purpose, we shall first show that if $N$ is a 2-step nilpotent Lie group endowed with a flat left-invariant Lorentz metric $g,$ then the restriction of $g$ to the center $Z$ of $N$ is degenerate. We shall then determine all 2-step nilpotent Lie groups that can admit a flat left-invariant Lorentz metric. We show that they are trivial central extensions of the three-dimensional Heisenberg Lie group $H_{3}$. If $\left ( N,g\right )$ is one such group, we prove that no timelike geodesic in $\left ( N,g\right )$ can be translated by an element of $N.$ By the way, we rediscover that the Heisenberg Lie group $H_{2k+1}$ admits a flat left-invariant Lorentz metric if and only if $k=1.$

References

A. Aubert, Structures affines et pseudo-métriques invariantes à gauche sur des groupes de Lie, Thesis, Univ. of Montpellier II, 1996.
John K. Beem, Paul E. Ehrlich, and Kevin L. Easley, Global Lorentzian geometry, 2nd ed., Monographs and Textbooks in Pure and Applied Mathematics, vol. 202, Marcel Dekker, Inc., New York, 1996. MR 1384756
Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Mathematical Library, Vol. 9, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0458335
L. A. Cordero and P. E. Parker, Pseudo-Riemannian 2-step nilpotent Lie groups, Preprint (1999).
Patrick Eberlein, Geometry of $2$-step nilpotent groups with a left invariant metric, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 5, 611–660. MR 1296558, DOI 10.24033/asens.1702
David Fried and William M. Goldman, Three-dimensional affine crystallographic groups, Adv. in Math. 47 (1983), no. 1, 1–49. MR 689763, DOI 10.1016/0001-8708(83)90053-1
David Fried, Flat spacetimes, J. Differential Geom. 26 (1987), no. 3, 385–396. MR 910014
M. Guediri, Sur la complétude des pseudo-métriques invariantes a gauche sur les groupes de Lie nilpotents, Rend. Sem. Mat. Univ. Politec. Torino 52 (1994), no. 4, 371–376 (French, with English summary). MR 1345607
M. Guediri, On the existence of closed timelike geodesics in compact spacetimes, Math. Z. 239 (2002), 277-291.
M. Guediri, Lorentz geometry of 2-step nilpotent Lie groups, (October 24, 2000) Submitted.
S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge Monographs on Mathematical Physics, No. 1, Cambridge University Press, London-New York, 1973. MR 0424186, DOI 10.1017/CBO9780511524646
Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
Aroldo Kaplan, Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. Amer. Math. Soc. 258 (1980), no. 1, 147–153. MR 554324, DOI 10.1090/S0002-9947-1980-0554324-X
Aroldo Kaplan, Riemannian nilmanifolds attached to Clifford modules, Geom. Dedicata 11 (1981), no. 2, 127–136. MR 621376, DOI 10.1007/BF00147615
Aroldo Kaplan, On the geometry of groups of Heisenberg type, Bull. London Math. Soc. 15 (1983), no. 1, 35–42. MR 686346, DOI 10.1112/blms/15.1.35
Katsumi Nomizu, Left-invariant Lorentz metrics on Lie groups, Osaka Math. J. 16 (1979), no. 1, 143–150. MR 527022
John F. Price, Lie groups and compact groups, London Mathematical Society Lecture Note Series, No. 25, Cambridge University Press, Cambridge-New York-Melbourne, 1977. MR 0450449, DOI 10.1017/CBO9780511600715
M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972. MR 0507234, DOI 10.1007/978-3-642-86426-1
Frank J. Tipler, Existence of closed timelike geodesics in Lorentz spaces, Proc. Amer. Math. Soc. 76 (1979), no. 1, 145–147. MR 534406, DOI 10.1090/S0002-9939-1979-0534406-6