Some topological properties of cohomogeneity one manifolds with negative curvature
The main results of this paper are the following: 1) Let \(M\) be a complete Riemannian manifold with negative sectional curvature and let \(G\) be a closed Lie group of isometries with regular orbits of codimension 1. Then only two possibilities can occur: a) \(G \) has no singular orbit and \(M\) is diffeomorphic to \(G/K \times \mathbb{R}\), where \(K\) is the isotropy subgroup of a point \(p \in M\); b) \(G\) admits exactly one singular orbit. 2) Under the same hypotheses, if \(G\) has a singular orbit, which does not coincide with the trace of a geodesic, then \(M\) is simply connected. 3) Under the same hypotheses, if \(M\) is not simply connected and of dimension \(n \geq 3\), then either each orbit of \(G\) is regular and diffeomorphic to \(T^k \times \mathbb{R}^q\), or there exists a 1-dimensional singular orbit and each regular orbit is covered by \(S^{n - 2} \times \mathbb{R}\). Other interesting results are obtained in case when the group \(G\) is supposed to be semisimple or compact.
scientific article; zbMATH DE number 854359