On the topological properties of some cohomogeneity one manifolds of some positive curvature
by
S. M.B. Kashani
Tarbiat Modarres University, Iran

Poster

In this paper we study some non positively curved (especially flat) Riemannian manifolds acted on by a closed Lie group of isometries with principal orbits of codimension one. Among other results, it is proved that the orbit space of flat cohomogeneity one manifolds can not be [0, 1] and the singular orbit (if it exists) is a totally geodesic submanifold.

(Lo que sigue lo escribe como anexco al abstract)

In this research our main results are:

1. Every homogeneous Riemannian manifold of nonpositive curvature is of cohomogeneity one under the action of a Lie group G, which is closed in the full isometry group.

2. Let M be a simply connected cohomogeneity one Riemannian manifold of nonpositive curvature, B a singular orbit and H=G_p the isotropy subgroup at a point p in B. Then H is maximal compact in G and there exists a solvable subgroup S of G, which acts transitively on B.

3. If M is a simply connected, cohomogeneity one Riemannian manifold of non positive curvature, then there is at most one singular orbit and if it exists, there is also a solvable subgroup S of G acting simply transitively on it.

4. Let M be a cohomogeneity one Riemannian manifold under the action of a closed Lie subgroup G of ISO (M) and with non positive curvature. Let [(\pi)\tilde]: [G\tilde] --> G be a finite sheeted covering space. Then M has at most one singular nonexceptional orbit. Hence, as a G-manifold it can be represented in one of the following forms.

a)

M=G ×_H V where V is an H-module with transitive action of H on the spheres (in other words M is a homogeneous vector bundle over G/H).

b)

M=G/H×R.

c)

M is a fibering over S¹ with a fiber G/H, here H is the maximal compact subgroup of G. In particular, if G is semisimple, G/H is a symmetric space of non positive curvature.

6. Suppose M is as described in 5 and that it is not simply connected. If n=dimM >= 3 and B is a singular orbit for the action of G, then B =~ T_l×R^m, l >= 1, \pi₁(M) =~ Z^l.
7. Let M be as described in 5 and that it is not simply connected and dimM >= 3, then either

a)

\Omega = M/G =~ R or S¹, each orbit is a solvmanifold and M is diffeomorphic to T_p×R^m, p, m >= 0, p+m=n-1. If M is compact, or if the orbits are simply connected, then M/G =~ S¹, in the second case \pi₁(M)=Z.

b)

\Omega = M/G =~ R₊, \pi₁(M) =~ Z^l, l >= 1. Moreover in case (a), if G is compact, then G is a torus and the action is free.

Date received: April 25, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-33.