Topological properties of some cohomogeneity one Riemannian manifolds of nonpositive curvature. (English) Zbl 1017.53038

Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 3rd international conference on geometry, integrability and quantization, Varna, Bulgaria, June 14-23, 2001. Sofia: Coral Press Scientific Publishing. 351-359 (2002).
The authors consider a complete Riemannian manifold \(M\) with nonpositive curvature and of cohomogeneity one under the action of a connected Lie group \(G\) which is closed in the full group of isometries of \(M\). In the first part of the paper they generalize some theorems by F. Podestà and A. Spiro [Ann. Global Anal. Geom. 14, 69-79 (1996; Zbl 0846.53024)] to the case when \(M\) is a so-called UND manifold, i.e. when \(M\) is universally and negatively decomposable (UND for short) in the sense that its universal covering manifold admits a suitable decomposition. It is proved that if \(M\) is a nonsimply connected UND cohomogeneity one Riemannian manifold then there is at most one singular orbit. Moreover, singular orbits must be non-exceptional. On the other hand, it is shown that such orbit is a totally geodesic submanifold.
In the second part the authors consider the case when \(M\) is flat and 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 \({\mathbb R}^k\times {\mathbb T}^m\).
For the entire collection see [Zbl 0980.00035].


53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions


Zbl 0846.53024