Homogeneous Riemannian manifolds and the visibility axiom
type: Item
instance of: scholarly article
title: Homogeneous Riemannian manifolds and the visibility axiom
zbMATH Open document ID: 0563.53039
DOI: 10.1007/BF00181590
author: Maria J. Druetta
published in: Geometriae Dedicata
publication date: 1985-01-01T00:00:00Z
review text: In this paper we study the Riemannian homogeneous spaces H of nonpositive sectional curvature satisfying the visibility axiom. If G is a simply transitive and solvable Lie group of isometries of H, the action of G in the points at infinity H(\(\infty)\) of H is completely described even when H has no focal points. We also describe the Lie groups that equipped with a left invariant metric of nonpositive curvature, satisfy the visibility axiom. As a consequence, we obtain a characterization of the Lie groups admitting left invariant metrics satisfying the visibility axiom. They are the same as those admitting left invariant metrics of negative curvature. Moreover, homogeneous visibility manifolds do not necessarily have negative curvature. We exhibit three-dimensional homogeneous visibility manifolds with some zero sectional curvature. Finally, a relationship between the algebraic structure of G and the geometric structure of H is obtained if visibility fails.
Mathematics Subject Classification ID: 53C30
zbMATH DE Number: 3897771
zbMATH Keywords: nonpositive sectional curvature
zbMATH Keywords: visibility axiom
zbMATH Keywords: simply transitive
zbMATH Keywords: solvable Lie group
zbMATH Keywords: homogeneous visibility manifolds
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00181590
OpenAlex ID: W1964910787
description: scientific article; zbMATH DE number 3897771