Geometric vector fields on Lie groups
type: Item
instance of: scholarly article
title: Geometric vector fields on Lie groups
zbMATH Open document ID: 0930.53025
DOI: 10.1016/S0926-2245(96)00049-6
author: Gerard Walschap
published in: Differential Geometry and its Applications
publication date: 2000-02-15T00:00:00Z
review text: On a Lie group \(G\) endowed with a left invariant metric, the author considers foliations consisting of left cosets of a connected Lie subgroup \(H\) such that the leaves are locally equidistant. In terms of the Lie algebras \(g\) and \(h\) of \(G\) and \(H\) respectively, the previous condition amounts to \(\langle[v,x],x\rangle=0\) for any \(v\in h\) and \(x\in h^\perp\). This kind of foliations are called ``left invariant''. They are called ``homogeneous'' when, locally, the leaves coincide with the orbits of some group of isometries acting freely on \(G\). If the foliation is one-dimensional, the vectors \(v\) in \(g\) giving rise to homogeneous flows are characterized in Proposition 2.1. Furthermore, in Theorem 2.4, it is shown that if \(G\) is unimodular, the directions producing homogeneous foliations have non-negative Ricci curvature. As a consequence, if \(v\) spans a one-dimensional ideal on a unimodular \(g\), the Ricci curvature in the direction of \(v\) is non-negative for any left invariant metric. Moreover, in Theorem 2.6, the author characterizes those directions on a Lie algebra \(g\) such that, for any left invariant metric, the Ricci curvature is non-negative. In Section 3, the author gives conditions on \(g\) and \(h\) in order to produce homogeneous foliations of higher dimensions with minimal leaves. Section 4 is devoted to the study of codimension one foliations in the Heisenberg group \(H_n\): it is shown that they are precisely the left invariants ones; they are generated by an ideal and have minimal leaves (Theorem 4.4).
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C30
zbMATH DE Number: 1145938
zbMATH Keywords: left invariant foliation
zbMATH Keywords: homogeneous foliation
zbMATH Keywords: non-negative Ricci curvature
zbMATH Keywords: Heisenberg group
Wikidata QID: Q115337592
reviewed by: Isabel Dotti Miatello
MaRDI profile type: Publication
cites work: A geometric property of central elements
cites work: Negative Ricci curvature on complex simple Lie groups
cites work: Comparison theorems and hypersurfaces
cites work: Q3681515
cites work: The low-dimensional metric foliations of Euclidean spheres
cites work: On homogeneous manifolds of negative curvature
cites work: Riemannian foliations on manifolds with non-negative curvature
cites work: Curvatures of left invariant metrics on Lie groups
cites work: The fundamental equations of a submersion
cites work: Minimal foliations on Lie groups
cites work: Metric foliations and curvature
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description: scientific article; zbMATH DE number 1145938