Orbit Spaces Arising from Isometric Actions on Hyperbolic Spaces

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