Harmonic endomorphism fields

Garcı́a-Rı́o, Eduardo; Vanhecke, Lieven; Abal, María Elena Vázquez

doi:10.1215/ijm/1255985842

Cited by 19 publications

(17 citation statements)

References 9 publications

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“…In the first part of this paper we show that the proper almost complex structure J on a 4-dimensional Walker manifold (W, q, D) (from [26]) is harmonic (in the sense of [21]) if and only if (J, q) is almost Kähler.…”

Section: Introductionmentioning

confidence: 94%

“…Of great interest in mathematics, harmonic maps, defined as critical points of the Dirichlet energy functional, were known before 1964, but research in this direction was undertaken by Eells and Sampson, who published their seminal work, [19], followed by a series of papers on the study of harmonic maps and morphisms (see The Atlas of Harmonic Morphisms, http://riemann.unica.it/ montaldo/homepage/atlas/). Moreover, harmonicity is studied in the context of sections in vector bundles, tensor fields, connections, and minimal distributions (see [7,21,29,32,33]).…”

Section: Introductionmentioning

confidence: 99%

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Harmonic functions and quadratic harmonic morphisms on Walker spaces

Bejan¹,

Druţă-Romaniuc²

2016

Turk J Math

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Let (W, q, D) be a 4-dimensional Walker manifold. After providing a characterization and some examples for several special (1, 1) -tensor fields on (W, q, D) , we prove that the proper almost complex structure J , introduced by Matsushita, is harmonic in the sense of García-Río et al. if and only if the almost Hermitian structure (J, q) is almost Kähler. We classify all harmonic functions locally defined on (W, q, D) . We deal with the harmonicity of quadratic maps defined on R 4 (endowed with a Walker metric q ) to the n -dimensional semi-Euclidean space of index r , and then between local charts of two 4-dimensional Walker manifolds. We obtain here the necessary and sufficient conditions under which these maps are harmonic, horizontally weakly conformal, or harmonic morphisms with respect to q .

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Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Harmonic functions and quadratic harmonic morphisms on Walker spaces

Bejan¹,

Druţă-Romaniuc²

2016

Turk J Math

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“…For a further interpretation of Jacobi fields along the identity, recall that a (0, 2)tensor on a Riemannian manifold (M, g) is called harmonic [29] if it has vanishing divergence.…”

Section: Proposition 8 a Vector Field V On A Compact Riemannian Manimentioning

confidence: 99%