Harmonic-Killing vector fields
type: Item
instance of: scholarly article
title: Harmonic-Killing vector fields
zbMATH Open document ID: 1044.53052
author: M. Trinidad Pérez
author: C. T. J. Dodson
author: M. E. Vázquez-Abal
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2004-02-03T00:00:00Z
review text: The authors introduce the notion of 1-harmonic-Killing (1-h-K) vector fields, i.e., vector fields whose corresponding 1-parameter group of local transformations consists of maps which have vanishing linear part of their tension field. It is shown that a vector field is a Jacobi field along the identity map if and only if it is a 1-h-K vector field. Examples of 1-h-K vector fields on (co-)Kähler manifolds are provided by the following result: for a given 1-h-K vector field on a (semi-)Riemannian manifold \((M,g)\) and a given parallel (1,1)-tensor field \(T\), the vector field \(TX\) is 1-h-K iff \(T\) commutes with the Ricci operator of \((M,g)\). Further, the authors introduce the notion of harmonic-Killing vector fields, i.e., of vector fields, whose 1-parameter group of local transformations consists of harmonic maps. These are characterized as Jacobi vector fields with harmonic flows. The relations between Killing, affine-Killing, conformal and harmonic-Killing vector fields are described, and the characterization of these kinds of vector fields with respect to their corresponding sections is provided.
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53C43
Mathematics Subject Classification ID: 53B20
zbMATH DE Number: 2031088
zbMATH Keywords: harmonic map
zbMATH Keywords: Killing vector field
zbMATH Keywords: Jacobi vector field
zbMATH Keywords: Ricci operator
reviewed by: Vladimir Balan
MaRDI profile type: Publication
Recommended article: Q5316740
Recommended article: On harmonic vector fields
Recommended article: Harmonic and minimal radial vector fields
Recommended article: Harmonic vector fields on space forms
Recommended article: Naturally Harmonic Vector Fields
Recommended article: Killing vector fields and harmonic superfield theories
Recommended article: Q4847030
Recommended article: Q3986606
Recommended article: Q4693507
Recommended article: Harmonic radial vector fields on harmonic spaces
description: scientific article; zbMATH DE number 2031088