Abstract
In this paper we study the cohomogeneity one de Sitter space S 1 n . We consider the actions in both proper and non-proper cases. In the first case we characterize the acting groups and orbits and we prove that the orbit space is homeomorphic to ℝ. In the latter case we determine the groups and consequently the orbits in some different cases and prove that the orbit space is not Hausdorff.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alekseevsky, A. V., Alekseevsky, D. V.: G-manifolds with one-dimensional orbit space. Adv. Sov. Math., 8, 1–31 (1992)
Berard-Bergery, L.: Sur de nouvells variété riemanniennes d’Einstein. Inst. Élie Cartan, 6, 1–60 (1982)
Podesta, F., Spiro, A.: Some topological properties of chomogeneity one manifolds with negative curvature. Ann. Global Anal. Geom., 14, 69–79 (1996)
Palais, R. S., Terng, CH. L.: A general theory of canonical forms. Trans. Amer. Math. Soc., 300, 771–789 (1987)
Searle, C.: Cohomogeneity and positive curvature in low dimension. Math. Z., 214, 491–498 (1993)
Verdiani, L.: Invariant metrics on cohomogeneity onemanifolds. Geom. Ded., 77, 77–111 (1999)
Ahmadi, P., Kashani, S. M. B.: Cohomogeneity one Minkowski space ℝ 1n . Preprint
Di Scala, A. J., Olmos, C.: The geometry of homogeneous submanifolds of hyperbolic space. Math. Z., 237, 199–209 (2001)
Alekseevsky, D. V.: On a proper action of a Lie group. Uspekhi Mat. Nauk, 34, 219–220 (1979)
Duistermaat, J. J., Kolk, J. A. C.: Lie Groups, Springer, Berlin, 2000
Bredon, G. E.: Introduction to Compact Transformation Groups, Academic Press, New York, 1972
Gorbatsevich, V. V., Onishik, A. L., Vinberg, E. B.: Lie groups and Lie algebras III (Structure of Lie Groups and Lie Algebras), Springer-Verlag, 1994
Adams, S.: Dynamics on Lorentz Manifolds, World Scientific, Hong Kong, 2001
Abedi, H., Alekseevsky, D., Kashani, S. M. B.: Cohomogeneity one Riemannian manifolds of nonpositive curvature. Diff. Geom. Appl., 25, 561–581 (2007)
Borel, H.: Some remarks about Lie groups transitive on sphere and tori. Bull. Amer. Math. Soc., 55, 580–587 (1949)
Knapp, A. W.: Representation Theory of Semisimple Groups, Princeton University Press, Princeton, 1986
Wolf, J. A.: Spaces of Constant Curvature, McGraw-Hill, New York, 1967
O’Neill, B.: Semi-Riemannian Geometry with Application to Relativity, Academic Press, New York, 1983
Hatcher, A.: Algebraic Topology, Cambridge University Press, Cambridge, 2002
Thomova, Z., Winternitz, P.: Maximal abelian subgroups of the isometry and conformal groups of Euclidean and Minkowski spaces. J. Phys. A, Math. Gen., 31, 1831–1858 (1998)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by Iranian Presidential Office (Grant No. 83211)
Rights and permissions
About this article
Cite this article
Ahmadi, P., Kashani, S.M.B. & Abedi, H. Cohomogeneity one de Sitter space S 1 n . Acta. Math. Sin.-English Ser. 26, 1915–1926 (2010). https://doi.org/10.1007/s10114-010-8142-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-010-8142-3