Some remarks on \(G\)-actions with sections
The author considers a complete Riemannian manifold \((M,g)\) and an action on this manifold by a closed Lie subgroup \(G\) of the full isometry group \(I(M,g)\), where the action admits a section, that is, admits a closed connected submanifold \(\Sigma\) which meets every \(G\)-orbit orthogonally. The author gives complete proofs of several results which do not appear in the literature. For example, she proves that the set of regular points in \(\Sigma\) both open and dense in \(\Sigma\), and proves that each section \(\Sigma\) is a totally geodesic submanifold. The author then imposes the additional hypothesis that \(M\) has nonpositive sectional curvatures and applies the Morse-theoretic methods of Bott to the study of \(M\). She shows that the action is variationally complete; relates strong Jacobi fields, strong focal points, and singular non-exceptional points of principal orbits; and studies Weyl chambers and the Weyl group defined by the action.
scientific article; zbMATH DE number 1544941
scientific article