Killing vector fields on tangent sphere bundles
type: Item
instance of: scholarly article
title: Killing vector fields on tangent sphere bundles
zbMATH Open document ID: 0907.53033
author: Tatsuo Konno
published in: Kodai Mathematical Journal
publication date: 1999-03-04T00:00:00Z
review text: Let \((TM,g_s)\) be the tangent bundle of a Riemannian manifold \((M,g)\) where \(g_S\) denotes the Sasaki metric and let \((T_\lambda M, \overline{g}_S)\) be the tangent sphere bundle, i.e., the set of elements \((x,u)\) where \(x\in M\) and \(u\in T_x M\), \(g(u,u)= \lambda^2\), endowed with the metric \(\overline{g}_S\) induced from \(g_S\). \(T_\lambda M\) is a hypersurface of \(TM\). S. Tanno has determined the general form of the Killing vector fields on \((TM, g_S)\). In the paper under review, the author uses this result to determine the general form of the Killing vector fields \(X\) on \((T_\lambda M, \overline{g}_S)\) which are fiber-preserving, i.e., those \(X\) such that the (local) isometries \(\psi_t\) generated by \(X\) map each fiber into a fiber. To do this, he proves that each fiber-preserving Killing vector field on \((T_\lambda M, \overline{g}_S)\) can be extended to a Killing vector field on \((TM,g_S)\) and that conversely, the restriction of every Killing vector field on \((TM,g_S)\) which is tangent to \(T_\lambda M\) is a fiber-preserving Killing vector field on \((T_\lambda M,\overline{g}_S)\). The author also provides examples of spaces of constant curvature \((M(c),g)\), with specific curvature \(c\), all of whose Killing vector fields on the unit tangent sphere bundle \((T_1 M, \overline{g}_S)\) are fiber-preserving. This result has already been extended by S. Tanno and the author to all values of \(c\), except for \(c\neq 1\). For this last case, the geodesic flow vector is a Killing vector field on \((T_1 M, \overline{g}_S)\) which is not fiber-preserving.
reviewed by: Lieven Vanhecke
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 1208196
zbMATH Keywords: tangent sphere bundle
zbMATH Keywords: Killing vector fields
zbMATH Keywords: spaces of constant curvature
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138043835
OpenAlex ID: W2146333696
cites work: Curvature of the Induced Riemannian Metric on the Tangent Bundle of a Riemannian Manifold.
cites work: On the differential geometry of tangent bundles of Riemannian manifolds
cites work: Killing vectors and geodesic flow vectors on tangent bundles.
DOI: 10.2996/KMJ/1138043835
description: scientific article; zbMATH DE number 1208196