Affine transformations and parallel lightlike vector fields on compact Lorentzian 3-manifolds

Boubel, Charles; Mounoud, Pierre

doi:10.1090/tran/6645

Affine transformations and parallel lightlike vector fields on compact Lorentzian 3-manifolds
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by Charles Boubel and Pierre Mounoud PDF

Trans. Amer. Math. Soc. 368 (2016), 2223-2262 Request permission

Abstract:

We describe the compact Lorentzian $3$-manifolds admitting a parallel lightlike vector field. The classification of compact Lorentzian $3$-manifolds admitting nonisometric affine diffeomorphisms follows, together with the complete description of these morphisms. Such a Lorentzian manifold is in some sense an equivariant deformation of a flat one.

References

L. Auslander and L. Markus, Flat Lorentz $3$-manifolds, Mem. Amer. Math. Soc. 30 (1959), 60. MR 131842
Helga Baum, Kordian Lärz, and Thomas Leistner, On the full holonomy group of Lorentzian manifolds, Math. Z. 277 (2014), no. 3-4, 797–828. MR 3229966, DOI 10.1007/s00209-014-1279-5
Alberto Candel and Lawrence Conlon, Foliations. I, Graduate Studies in Mathematics, vol. 23, American Mathematical Society, Providence, RI, 2000. MR 1732868, DOI 10.1090/gsm/023
Yves Carrière, Flots riemanniens, Astérisque 116 (1984), 31–52 (French). Transversal structure of foliations (Toulouse, 1982). MR 755161
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
Bassam Fayad, Anatole Katok, and Alistar Windsor, Mixed spectrum reparameterizations of linear flows on ${\Bbb T}^2$, Mosc. Math. J. 1 (2001), no. 4, 521–537, 644 (English, with English and Russian summaries). Dedicated to the memory of I. G. Petrovskii on the occasion of his 100th anniversary. MR 1901073, DOI 10.17323/1609-4514-2001-1-4-521-537
Jacqueline Ferrand, The action of conformal transformations on a Riemannian manifold, Math. Ann. 304 (1996), no. 2, 277–291. MR 1371767, DOI 10.1007/BF01446294
Anton Galaev and Thomas Leistner, Holonomy groups of Lorentzian manifolds: classification, examples, and applications, Recent developments in pseudo-Riemannian geometry, ESI Lect. Math. Phys., Eur. Math. Soc., Zürich, 2008, pp. 53–96. MR 2436228, DOI 10.4171/051-1/2
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
Michael Gromov, Rigid transformations groups, Géométrie différentielle (Paris, 1986) Travaux en Cours, vol. 33, Hermann, Paris, 1988, pp. 65–139. MR 955852
Anatole Katok, Cocycles, cohomology and combinatorial constructions in ergodic theory, Smooth ergodic theory and its applications (Seattle, WA, 1999) Proc. Sympos. Pure Math., vol. 69, Amer. Math. Soc., Providence, RI, 2001, pp. 107–173. In collaboration with E. A. Robinson, Jr. MR 1858535, DOI 10.1090/pspum/069/1858535
A. N. Kolmogorov, On dynamical systems with an integral invariant on the torus, Doklady Akad. Nauk SSSR (N.S.) 93 (1953), 763–766 (Russian). MR 0062892
K. Lärz, Global aspects of holonomy in pseudo-Riemannian geometry. PhD thesis, Humboldt-Universität zu Berlin, 2011. http://edoc.hu-berlin.de/docviews/abstract.php?lang =ger&id=38481
T. Leistner and D. Schliebner, Completeness of compact Lorentzian manifolds with special holonomy, arXiv:1306.0120v2, 2013.
G. K. Martin and G. Thompson, Nonuniqueness of the metric in Lorentzian manifolds, Pacific J. Math. 158 (1993), no. 1, 177–187. MR 1200834
Vladimir S. Matveev, Proof of the projective Lichnerowicz-Obata conjecture, J. Differential Geom. 75 (2007), no. 3, 459–502. MR 2301453
Richard S. Palais, A global formulation of the Lie theory of transformation groups, Mem. Amer. Math. Soc. 22 (1957), iii+123. MR 121424
D. Schliebner, On the full holonomy of Lorentzian manifolds with parallel Weyl tensor, arxiv:1204.5907, 2012, revised 2014.
D. Schliebner, On Lorentzian manifolds with highest first Betti number, arXiv:1311.6723v1, 2013. To appear in Ann. Inst. Fourier (Grenoble).
H. Wu, Holonomy groups of indefinite metrics, Pacific J. Math. 20 (1967), 351–392. MR 212740
Kentaro Yano, On harmonic and Killing vector fields, Ann. of Math. (2) 55 (1952), 38–45. MR 46122, DOI 10.2307/1969418
Abdelghani Zeghib, Le groupe affine d’une variété riemannienne compacte, Comm. Anal. Geom. 5 (1997), no. 1, 199–211 (French). MR 1456311, DOI 10.4310/CAG.1997.v5.n1.a4
Abdelghani Zeghib, Killing fields in compact Lorentz $3$-manifolds, J. Differential Geom. 43 (1996), no. 4, 859–894. MR 1412688
A. Zeghib, On discrete transformation groups of Riemannian manifolds, preprint.