Orbit Spaces Arising from Isometric Actions on Hyperbolic Spaces

Authors
Reza Mirzaie
mojtaba heydari
imam KIhomeini International University
10.29252/mmr.3.2.147
Abstract
Let be a differentiable action of a Lie group on a differentiable manifold and consider the orbit space with the quotient topology. Dimension of is called the cohomogeneity of the action of on . If is a differentiable manifold of cohomogeneity one under the action of a compact and connected Lie group, then the orbit space is homeomorphic to one of the spaces , , or . In this paper we suppose that the hyperbolic space is of cohomogeneity two under the action of , a connected and closed subgroup of Then we prove that its orbit space is homeomorphic to or Also we prove that either all orbits are diffeomorphic to or there are nonnegative integers such that some orbits are diffeomorphic to , and the other orbits are diffeomorphic to , where may be a sphere, a homogeneous hypersurface of sphere or a helix in some Euclidean space.

../files/site1/files/0Abstract6.pdf
Keywords
Manifold
Hyperbolic space
Cohomogeneity
Orbit space
Isometry
1. Arouche A., Deffaf M., Zeghib A., "On lorentze dynamics: from group actions to warped products via homogeneous spaces", Trans. Am. Math. Soc, 359 (2007) 1253-1263. 2. Boubel C., Zeghib A., "Isometric actions of Lie subgroups of the Moebius group", Nonlinearity 17 (2004) 1677-1688. 3. Bredon. G. E., "Introduction to compact transformation groups", Acad. Press, New York, London (1972). 4. Di Scala A. J., Olmos C., "The geometry of homogeneous submanifolds of hyperbolic space", Math. Z, 237 (2001) 199-209. 5. Eberlin P., O’Neil B., "Visibility manifolds", Pasific J. Math, 46, 1 (1973) 45-109. 6. Hsiang W. Y., Lawson H. B., "Minimal submanifolds of low Cohomogeneity", J. Diff. Geometry, 5 (1971) 1-38. 7. Michor P. W., "Isometric actions of Lie groups and invariants", Lecture course at the university of Vienna (1996) 97. 8. Mirzaie R.," Cohomogeneity two actions on flat Riemannian manifolds", Acta Mathematica Sinica, 23 (2007) 1587-1592. 9. Mirzaie R., "On negatively curved Riemannian manifold of low cohomogeneity", Hokkaido mathematical journal, 38 (2009) 797-803. 10. Mirzaie R., "On orbits of isometric actions on flat Riemannian manifolds", Kyushu J. Math, 65 (2011) 383-393. 11. Mirzaie R., "On Riemannian manifolds of constant negative curvature", J. Korean Math. Soc, 48 (2011) 23-31. 12. Mirzaie R., "On topology of some Riemannian manifolds of negative curvature with a compact Lie group of isometries", Hokkaido Math. J. (2015). 13. Mirzaie R., "On Euclidean G-manifolds which have two dimensional orbit spaces", International J. Math, 22, 3 (2011) 399-406. 14. Mirzaie R., Kashani S. M. B., "On cohomogeneity one flat Riemannian manifolds", GlasgwMath. J, 44 (2002) 185-190. 15. Mostert P., "On a compact Lie group action on manifolds", Ann. Math, 65 (1957) 447. 16. Palais R. S., Terrg CH. L., "A general theory of canonical forms", Trans. Am. Math.Soc, 300 (1987) 771-789. 17. Podesta F., Spiro A., "Some topological properties of cohomogeneity one manifolds with negative curvature" Ann. Global. Anal. Geom, 14 (1966) 69-79. 18. Searle C., "Cohomogeneity and positive curvature in low dimensions", Math. Z, 214 (1993) 491-498. 19. Straume E., "Compact connected Lie transformation groups on spheres with low cohomogeneity. I", Memoris of the AMS, Vol. 119, 569 (1996). 20. Straume E., "Compact connected Lie transformation groups on spheres with low cohomogeneity. II", Memoirs of the AMS, Vol. 125, 595 (1997) 
Mathematical Research
Volume 3, Issue 3 - Serial Number 7
Autumn 2017
Pages 147-154