## Proceedings of the International Conference on Geometry, Integrability and Quantization

### Topological Properties of Some Cohomogeneity One Riemannian Manifolds of Nonpositive Curvature

#### Abstract

In this paper we study some non-positively curved Riemannian manifolds acted on by a Lie group of isometries with principal orbits of codimension one. Among other results it is proved that if the universal covering manifold satisfies some conditions then every non-exceptional singular orbit is a totally geodesic submanifold. When $M$ is flat and is not toruslike, it is proved that either each orbit is isometric to $\mathbb{R}^k \times \mathbb{T}^m$ or there is a singular orbit. If the singular orbit is unique and non-exceptional, then it is isometric to $\mathbb{R}^k \times \mathbb{T}^m$.

#### Article information

Dates
First available in Project Euclid: 12 June 2015

Permanent link to this document
https://projecteuclid.org/ euclid.pgiq/1434145481

Digital Object Identifier
doi:10.7546/giq-3-2002-351-359

Mathematical Reviews number (MathSciNet)
MR1884859

Zentralblatt MATH identifier
1017.53038

#### Citation

Mirzaie, R.; Kashani, S. Topological Properties of Some Cohomogeneity One Riemannian Manifolds of Nonpositive Curvature. Proceedings of the Third International Conference on Geometry, Integrability and Quantization, 351--359, Coral Press Scientific Publishing, Sofia, Bulgaria, 2002. doi:10.7546/giq-3-2002-351-359. https://projecteuclid.org/euclid.pgiq/1434145481