Lorentzian G-manifolds of Constant Positive Curvature

Year 2024, Volume: 17 Issue: 2, 551 - 558

Reza Mirzaie , Jafar Ebrahimi

https://doi.org/10.36890/iejg.1405824

Abstract

We study the orbits arising from isometric actions of connected Lie groups on Lorentzian manifolds
with constant positive curvature.

Keywords

Lorentzian manifold, Lorentzian space, timelike plane

References

• [1] Abedi, H., Kashani, S. M. B.: Cohomogeneity one Riemannian manifolds of constant positive curvature. Journal of the Korean Math. Soci. 44, 799-807 (2007). https://doi.org/10.1016/j.difgeo.2007.06.006
• [2] Ahmadi, P., Kashani, S. M. B.: Cohomogeneity one Anti De Sitter space H3 1 . Bulletin of the Irannian Math. Soc. 35, 221-233 (2009).
• [3] Ahmadi, P.: Cohomogeneity one Lorentzian space forms: Minkowski, de Sitter and anti de Sitter spaces. Lap Lambert Academic Publishing (2011).
• [4] Alekseevsky, D. V.: Homogeneous Lorentzian manifolds of a semisimple group. Journal of Geomerty and Physics. 62, 631-645 (2012). https://doi.org/10.48550/arxiv.1101.3093
• [5] Bredon, G. E.: Introdution to compact transformation groups. Acad. Press. New york, London (1972).
• [6] Berndt, J., Bruk, M.: Cohomogeneity one actions on hyperbolic spaces. Journal fur die Reine und Angewandte Mathematik. 209-235 (2001). https://doi.org/10.1515/crll.2001.093
• [7] Brendt, J., Console, S., Olmos, C.: Submanifolds and holonomy, Chapman and Hall/CRS. London, New york (2003).
• [8] Calvaruso, G., Lopez, C.: Pseudo-Riemannian homogeneous structures. Springer Cham, Switzerland (2019).
• [9] Dadok, J.: Polar coordinates induced by actions of compact Lie groups. Trans. Amer. Math. Soc. 228, 125-137 (1985).
• [10] Diaz-Ramos, J. C., Kashani, S. M. B., Vanaei, M. J.: Cohomogeneity one actions on anti de Sitter spacetimes. Preprint arxiv:1609.05644[math.DG]. https://doi.org/10.48550/arXiv.1609.05644
• [11] Eberlin, P., O’Neil, B.:Visibility manifolds. Pasific J. Math. 46, 45-109 (1973).
• [12] Kobayashi, S.: Homogeneous Riemannian manifolds of negative curuature. Tohoku Math. J. 14, 413-415 (1962). https://doi.org/10.2748/tmj/1178244077
• [13] Kollros, A.: A classification of hyperpolar and cohomogeneity one actions. Trans. Amer. Math. Soc. 354, 571-612 (2002).
• [14] Mirzaie, R.: Topological properties of some flat Lorentzian manifolds of low cohomogeneity. Hiroshima Math. J. 44, 267-274 (2014). https://doi.org/10.32917/hmj/1419619747
• [15] Mirzaie, R.: Orbit space of cohomogeneity two flat Riemannian manifolds. Balkan Journal of Geometry and Its Applications. 2, 25-33 (2018).
• [16] Mostert, P.: On a compact Lie group action on manifolds. Ann. Math. 65, 447-455 (1957).
• [17] O’Neill, B.: Semi-Riemannian geometry with applications to relativity. Academic Press. New York (1983).
• [18] Patrangenaru, V.: Lorentzian manifolds with the three largest degrees of symmetry. Geometria Dedicata. 102, 25-33 (2003). https://doi.org/10.1023/b:geom.0000006588.95481.1c
• [19] Podesta, F., Spiro, A.: Some topological properties of cohomogeneity one manifolds with negative curvature. Ann. Glob. Anal. Geom. 14, 69-79 (1966). https://doi.org/10.1007/bf00128196
• [20] Podesta, F., Verdiani, L.: Positively curved 7-dimensional manifolds. Quart. J. Math. Oxford Ser. 50, 497-504 (1999). https://doi.org/10.48550/arxiv.dg-ga/9712002
• [21] Searle, C.: Cohomogeneity and positive curvature in low dimensions. Math. Z. 214, 491-498 (1993).
• [22] Scala, A. J. Di., Olmos, C.: The geometry of homogeneous submanifolds of hyperbolic space. Math. Z. 237, 199-209 (2001). https://doi.org/10.1007/PL00004860
• [23] Szeghy, D.: Isometric actions of compact connected Lie groups on globally hyperbolic Lorentz manifolds. Publ. Math. Debrecen. 71, 229-243 (2007). https://doi.org/10.5486/pmd.2007.3730
• [24] Wolf, J. A.: Flat homogeneous pseudo-Riemannian manifolds. Geometria Dedicata. 57, 111-120 (1995). https://doi.org/10.1007/BF01264064
• [25] Wolf, J. A.: Spaces of constant curvature. McGraw-Hill, New York (1967).