Some remarks on \(G\)-actions with sections
type: Item
instance of: scholarly article
title: Some remarks on \(G\)-actions with sections
zbMATH Open document ID: 0994.53017
author: Elisa Cage Casini
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2002-10-10T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108517
review text: 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.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 57S15
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 1544941
zbMATH Keywords: \(G\)-orbit
zbMATH Keywords: set of regular points
zbMATH Keywords: totally geodesic submanifold
zbMATH Keywords: nonpositive sectional curvatures
zbMATH Keywords: variationally complete
zbMATH Keywords: principal orbits
zbMATH Keywords: Weyl chambers
zbMATH Keywords: Weyl group
reviewed by: James W. Cannon
MaRDI profile type: Publication
cites work: Q4370979
cites work: An application of the Morse theory to the topology of Lie-groups
cites work: Applications of the Theory of Morse to Symmetric Spaces
cites work: Q5658847
cites work: Variational completeness and \(K\)-transversal domains
cites work: Q4741934
cites work: Q4303029
cites work: Q5731354
cites work: A General Theory of Canonical Forms
cites work: Critical point theory and submanifold geometry
cites work: Q3943618
cites work: Orthogonally transversal submanifolds and the generalizations of the Weyl group
description: scientific article; zbMATH DE number 1544941