Harmonic sections of homogeneous fibre bundles.
type: Item
instance of: scholarly article
title: Harmonic sections of homogeneous fibre bundles.
zbMATH Open document ID: 1058.53053
DOI: 10.1016/S0926-2245(03)00021-4
published in: Differential Geometry and its Applications
publication date: 2003-10-14T00:00:00Z
review text: Let \(\pi : N \rightarrow M\) be a smooth reductive homogeneous fibre bundle equipped with a Kaluza-Klein geometric structure. Let \(\sigma : M \rightarrow N\) be a smooth section. One defines the super-curvature of \(\sigma\) to be the tensor field \(\alpha^v\sigma = \nabla^v \text{ d}^v \sigma\) and the vertical tension field of \(\sigma\) to be \(\tau^v \sigma = \text{ Tr} \nabla^v \text{ d}^v \sigma\), where \(\text{ d}^v\) is the vertical component of the exterior differential, and \(\nabla^v\) is the vertical component of the Levi-Civita connection of \(N\). The section \(\sigma\) is defined to be harmonic if \(\tau^v\sigma=0\) and is super-flat if \(\alpha^v \sigma = 0\). This paper establishes the following sequence of implications for sections of \(N\): \[ \text{super-flat} \Rightarrow \text{ totally\;geodesic} \Rightarrow \text{ harmonic\;map} \Rightarrow \text{ harmonic\;section}. \] For flat bundles, or more generally flat sections, there are the equivalencies: super-flat \(\Leftrightarrow\) totally geodesic, and harmonic map \(\Leftrightarrow\) harmonic section. The author constructs examples to show these can fail to be equivalent for nonflat bundles.
Mathematics Subject Classification ID: 53C43
Mathematics Subject Classification ID: 58E15
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 1992875
zbMATH Keywords: harmonic section
zbMATH Keywords: Super-curvature
zbMATH Keywords: Super-flat
zbMATH Keywords: Twistor bundle
zbMATH Keywords: Almost-Hermitian
Wikidata QID: Q115338032
author: C. M. Wood
reviewed by: James Hebda
MaRDI profile type: Publication
cites work: Q4720334
cites work: Q4532261
cites work: On the energy of unit vector fields with isolated singularities
cites work: Twistor theory for Riemannian symmetric spaces. With applications to harmonic maps of Riemann surfaces
cites work: Harmonic almost-complex structures on twistor spaces
cites work: On the Geometry of the Tangent Bundle.
cites work: Energy and volume of unit vector fields on three-dimensional Riemannian manifolds.
cites work: Q5699227
cites work: Nearly Kähler manifolds
cites work: Riemannian manifolds with geodesic symmetries of order 3
cites work: Unit vector fields on spheres, which are harmonic maps
cites work: A Sufficient Condition that a Mapping of Riemannian Manifolds be a Fibre Bundle
cites work: The energy of unit vector fields on the 3-sphere
cites work: Q5731354
cites work: On sections of fiber bundles which are harmonic maps
cites work: Curvature of the Induced Riemannian Metric on the Tangent Bundle of a Riemannian Manifold.
cites work: The fundamental equations of a submersion
cites work: Q3346969
cites work: Q3721461
cites work: Minimality and harmonicity for Hopf vector fields
cites work: Totally geodesic maps
cites work: Q4854053
cites work: Harmonic sections and equivariant harmonic maps
full work available at URL: s0926-2245(03)00021-4
OpenAlex ID: W2041476095
description: scientific article; zbMATH DE number 1992875