Harmonic vector fields. Variational principles and differential geometry
type: Item
instance of: scholarly article
zbMATH Open document ID: 1245.53002
author: Sorin Dragomir
author: Domenico Perrone
publication date: 2012-05-18T00:00:00Z
Mathematics Subject Classification ID: 53-02
Mathematics Subject Classification ID: 53-01
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 53B20
Mathematics Subject Classification ID: 53B35
Mathematics Subject Classification ID: 53D10
Mathematics Subject Classification ID: 32V20
Mathematics Subject Classification ID: 53C43
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 6036200
zbMATH Keywords: differential geometry
zbMATH Keywords: manifold
zbMATH Keywords: vector field
zbMATH Keywords: harmonic vector field
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: harmonic map
zbMATH Keywords: variational principle
zbMATH Keywords: contact manifold
zbMATH Keywords: H-contact manifold
zbMATH Keywords: CR manifold
zbMATH Keywords: Sasaki metric
zbMATH Keywords: Hopf vector field
zbMATH Keywords: Killing vector field
zbMATH Keywords: Reeb vector field
zbMATH Keywords: tangent sphere bundle
zbMATH Keywords: critical point
zbMATH Keywords: Riemannian torus
zbMATH Keywords: Stokes' theorem
zbMATH Keywords: complex Monge-Ampère equation
zbMATH Keywords: Reilly's formula
zbMATH Keywords: semi-Riemannian manifold
zbMATH Keywords: Lorentzian manifold
MaRDI profile type: Publication
title: Harmonic vector fields. Variational principles and differential geometry
review text: The term `harmonic vector field' is currently adopted in the literature with at least two different meanings, the simpler being a vector field that is locally the gradient of some harmonic (scalar) function. The harmonic vector fields considered in this book arise instead in connection with the theory of harmonic maps between Riemannian manifolds. If indeed a vector field \(X\) on a Riemannian manifold \(M\) is viewed (as usual) as a map \(M\to TM\), and giving \(TM\) the Sasaki metric, one could wonder when it becomes harmonic. In the case when \(M\) is compact and oriented, the answer has been known for a long time: \(X\) is a harmonic map \(M\to TM\) if and only if \(X\) is a parallel vector field on the Riemannian manifold \(M\), i.e., it is parallel along every path see [\textit{O. Nouhaud}, C. R. Acad. Sci., Paris, Sér. A 284, 815--818 (1977; Zbl 0349.53015)]. Hence, under such strict conditions there would be little to say (at least in the compact oriented case). However, there are weaker harmonicity conditions that are well worth taking into consideration. For instance, the Sasaki metric could be replaced by some of the so-called \(g\)-natural metrics (\(g\) being the metric on \(M\)). To state the `appropriate' harmonicity notion, the authors assume a variational viewpoint and, like in the theory of harmonic maps, the Dirichlet energy functional is considered. Then, the search for the appropriate notion may be reduced to the search for the appropriate functional space from which critical points should be taken. It turns out to be the space \(\Gamma^\infty\left(S(M)\right)\) of unit vector fields. The resulting theory is supported by a considerable amount of work by many authors for (roughly) the past two decades, of which the book gives a careful account.NEWLINENEWLINEAccording to the book's preface, the text is `mostly confined to the study of differential geometric property of harmonic vector fields' and of their geometric background, mainly within contac…
reviewed by: Alessandro De Paris
description: scientific article; zbMATH DE number 6036200