Sub-Lorentzian geometry on anti-de Sitter space. (English) Zbl 1151.53025
The authors give the precise form of left-invariant vector fields defining sub-Riemannian and sub-Lorentzian structures on anti-de Sitter group. In sections 3 and 4 the question of existence of smooth horizontal curves in the sub-Lorentzian manifold is studied. The Lagrangian and Hamiltonian formalisms are applied to find sub-Lorentzian geodesics in sections 5 and 6. Section 7 is devoted to the study of a sub-Riemannian geometry defined by the distribution generated by space-like vector fields of anti-de Sitter space. In both sub-Riemannian and sub-Lorentzian cases they find geodesics explicitely.
Reviewer: Jan Kurek (Lublin)
MSC:
| 53C17 | Sub-Riemannian geometry |
| 53C50 | Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics |
| 53C27 | Spin and Spin\({}^c\) geometry |
| 83C65 | Methods of noncommutative geometry in general relativity |
References:
| [1] | Bengtsson, I.; Sandin, P., Anti-de Sitter space, squashed and stretched, Classical Quantum Gravity, 23, 3, 971-986 (2006) · Zbl 1087.83058 |
| [2] | Beals, R.; Gaveau, B.; Greiner, P. C., Complex Hamiltonian mechanics and parametrices for subelliptic Laplacians, I, II, III, Bull. Sci. Math., 21, 1-3, 1-36 (1997), 97-149, 195-259 · Zbl 0886.35001 |
| [3] | Calin, O.; Chang, D. C.; Greiner, P. C., Geometric Analysis on the Heisenberg Group and Its Generalizations, AMS/IP Ser. Adv. Math., vol. 40 (2007), International Press: International Press Cambridge, MA · Zbl 1132.53001 |
| [4] | O. Calin, D.C. Chang, I. Markina, Sub-Riemannian geometry of the sphere \(S^3\), Canadian J. Math. (2008), in press; O. Calin, D.C. Chang, I. Markina, Sub-Riemannian geometry of the sphere \(S^3\), Canadian J. Math. (2008), in press |
| [5] | Carlip, S., Conformal field theory, \((2 + 1)\)-dimensional gravity and the BTZ black hole, Classical Quantum Gravity, 22, 12, R85-R123 (2005) · Zbl 1098.83001 |
| [6] | Chow, W. L., Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann., 117, 98-105 (1939) · Zbl 0022.02304 |
| [7] | Grochowski, M., Geodesics in the sub-Lorentzian geometry, Bull. Polish Acad. Sci. Math., 50, 2, 161-178 (2002) · Zbl 1043.53034 |
| [8] | Grochowski, M., On the Heisenberg sub-Lorentzian Metric on \(R^3\), Geometric Singularity Theory, vol. 65 (2004), Banach Center Publ., Polish Acad. Sci., pp. 57-65 · Zbl 1065.53055 |
| [9] | Hawking, S. W.; Ellis, G. F., The Large Scale Structure of Space Time (1973), Cambridge Univ. Press · Zbl 0265.53054 |
| [10] | Lu, Q. K., Heisenberg group and energy-momentum conservative law in de-Sitter spaces, Commun. Theor. Phys. (Beijing), 44, 3, 389-392 (2005) |
| [11] | Montgomery, R., A Tour of Subriemannian Geometries, Their Geodesics and Applications, Mathematical Surveys and Monographs, vol. 91 (2002), American Mathematical Society: American Mathematical Society Providence, RI, 259 pp. · Zbl 1044.53022 |
| [12] | Naber, G. L., The Geometry of Minkowski Spacetime. An Introduction to the Mathematics of the Special Theory of Relativity, Applied Mathematical Sciences, vol. 92 (1992), Springer-Verlag: Springer-Verlag New York, 257 pp. · Zbl 0757.53046 |
| [13] | Natrio, J., Relativity and singularities—a short introduction for mathematicians, Resenhas, 6, 4, 309-335 (2005) · Zbl 1391.53082 |
| [14] | O’Neill, B., Semi-Riemannian Geometry. With Applications to Relativity, Pure and Applied Mathematics, vol. 103 (1983), Academic Press, Inc.: Academic Press, Inc. New York · Zbl 0531.53051 |
| [15] | Oliva, W. M., Geometric Mechanics, Lecture Notes in Mathematics, vol. 1798 (2002), Springer-Verlag: Springer-Verlag Berlin, 270 pp. · Zbl 1013.70013 |
| [16] | Penrose, R., The Road to Reality. A Complete Guide to the Laws of the Universe (2005), Alfred A. Knopf, Inc.: Alfred A. Knopf, Inc. New York, 1099 pp. · Zbl 1188.00007 |
| [17] | Porteous, I. R., Clifford Algebras and the Classical Groups, Cambridge Studies in Advanced Mathematics, vol. 50 (1995), Cambridge University Press: Cambridge University Press Cambridge, 295 pp. · Zbl 0855.15019 |
| [18] | da Rocha, R.; de Oliveira, E. C., AdS geometry, projective embedded coordinates and associated isometry groups, Internat. J. Theoret. Phys., 45, 3, 575-588 (2006) · Zbl 1113.53047 |
| [19] | Strichartz, R. S., Sub-Riemannian geometry, J. Differential Geom., 24, 2, 221-263 (1986) · Zbl 0609.53021 |
| [20] | Strichartz, R. S., Corrections to: “Sub-Riemannian geometry”, J. Differential Geom.. J. Differential Geom., J. Differential Geom., 30, 2, 595-596 (1989) |
| [21] | (Bellaïche, A.; Risler, J.-J., Sub-Riemannian Geometry. Sub-Riemannian Geometry, Progress in Mathematics, vol. 144 (1996), Birkhäuser Verlag: Birkhäuser Verlag Basel), 393 pp. · Zbl 0862.53031 |
| [22] | Szekeres, P., A Course in Modern Mathematical Physics. Groups, Hilbert Space and Differential Geometry (2004), Cambridge University Press: Cambridge University Press Cambridge, 600 pp. · Zbl 1070.00002 |
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