Harmonicity of vector fields on four-dimensional generalized symmetric spaces
type: Item
instance of: scholarly article
title: Harmonicity of vector fields on four-dimensional generalized symmetric spaces
zbMATH Open document ID: 1246.53083
author: Giovanni Calvaruso
published in: Central European Journal of Mathematics
publication date: 2012-05-31T00:00:00Z
review text: Let \((M=G/H,g)\) denote a four-dimensional pseudo-Riemannian generalized symmetric space and \(\mathfrak{g}=\mathfrak{m}+\mathfrak{h}\) the corresponding decomposition of the Lie algebra \(\mathfrak{g}\) of \(G\). Let \(g^s\) be the Sasaki metric on \(TM\). Given a smooth vector field \(V\) on \(M\), the energy of \(V\) as the energy of the map \(V:(M,g)\to (TM,g^s)\), i.e. \[ E(V)=\frac 12\int_M(\text{tr}_gV^\ast g^s)\text{d}v=\frac n2\text{vol}(M,g)+\frac 12\int_M||\nabla V||^2\text{d}v. \] If \(V:(M,g)\to (TM,g^s)\) is a critical point for the energy functional, then \(V\) is said to define a harmonic map. A smooth vector field \(V\) is said to be a harmonic section if and only if it is a critical point of \(\displaystyle E^v=\frac 12\int_M||\nabla V||^2\text{d}v\). In this paper the author completely determine the harmonicity properties of vector fields belonging to \(\mathfrak{m}\). In some cases, all these vector fields are critical points for the energy functional restricted to vector fields. Vector fields defining harmonic maps are also classified. For the classification of generalized symmetric spaces, see the related paper [\textit{C. J. Kowalski} and \textit{J. Cerny}, Tensor, New Ser. 38, 255--267 (1982; Zbl 0505.53024)].
reviewed by: Weihuan Chen
Mathematics Subject Classification ID: 53C43
Mathematics Subject Classification ID: 53C50
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 58E20
zbMATH DE Number: 6039959
zbMATH Keywords: harmonic vector field
zbMATH Keywords: harmonic map
zbMATH Keywords: generalized symmetric spaces
zbMATH Keywords: pseudo-Riemannian space
MaRDI profile type: Publication
full work available at URL: s11533-011-0109-9
OpenAlex ID: W2032789296
Wikidata QID: Q126077740
cites work: Harmonicity of unit vector fields with respect to Riemannian \(g\)-natural metrics
cites work: Some examples of harmonic maps for \(g\)-natural metrics
cites work: Q3083015
cites work: HARMONIC SECTIONS OF TANGENT BUNDLES EQUIPPED WITH RIEMANNIAN g-NATURAL METRICS
cites work: Local rigidity of certain classes of almost Kähler 4-manifolds
cites work: Harmonic sections of Riemannian vector bundles, and metrics of Cheeger-Gromoll type
cites work: Harmonicity properties of invariant vector fields on three-dimensional Lorentzian Lie groups
cites work: Curvature properties of four-dimensional generalized symmetric spaces
cites work: Q4816553
cites work: Local rigidity and nonrigidity of symplectic pairs
cites work: Q4741953
cites work: Curvature properties of four-dimensional Walker metrics
cites work: Three-dimensional Lorentz manifolds admitting a parallel null vector field
cites work: Q3653694
cites work: Totally geodesic hypersurfaces of four-dimensional generalized symmetric spaces
cites work: Relationship between volume and energy of vector fields.
cites work: Q5465266
cites work: Harmonic and minimal invariant unit vector fields on homogeneous Riemannian manifolds
cites work: Spacelike energy of timelike unit vector fields on a Lorentzian manifold
cites work: Volume, energy and generalized energy of unit vector fields on Berger spheres: stability of Hopf vector fields
cites work: Unit vector fields on spheres, which are harmonic maps
cites work: Q3862702
cites work: Einstein-Maxwell equation on four-dimensional homogeneous spaces
cites work: On formality of generalized symmetric spaces
cites work: Generalized symmetric spaces
cites work: Q4119760
cites work: Q3312869
cites work: Q4800825
cites work: Pontryagin classes of generalized symmetric spaces
cites work: Total bending of vector fields on Riemannian manifolds
cites work: On the energy of a unit vector field
DOI: 10.2478/S11533-011-0109-9
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description: scientific article; zbMATH DE number 6039959