Orthogonal almost-complex structures of minimal energy

Bor, Gil; Hernández-Lamoneda, Luis; Salvai, Marcos

doi:10.1007/s10711-007-9160-x

Cited by 15 publications

(28 citation statements)

References 15 publications

(22 reference statements)

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“…Therefore it describes a sort of gauge theory on M what we call a Yang-Mills-Higgs-Nijenhuis field theory. The peculiarities here compared to the usual Yang-Mills-Higgs Lagrangian are that the curvature has again been replaced with the torsion, the connection is minimally coupled to the metric but its usual minimal coupling ∇Φ to the Higgs field has been replaced with N ∇ Φ regarded as a non-minimal quadratic coupling in (2). This further departure from the physical side gives rise again to a familiar structure on M on the mathematical side as follows.…”

Section: Geometric Structures and Symmetry Breakingmentioning

confidence: 99%

Complex structure on the six dimensional sphere from a spontaneous symmetry breaking

Etesi

2015

Journal of Mathematical Physics

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Existence of a complex structure on the 6 dimensional sphere is proved in this paper. The proof is based on re-interpreting a hypothetical complex structure as a classical ground state of a Yang-Mills-Higgs-like theory on S 6 . This classical vacuum solution is then constructed by Fourier expansion (dimensional reduction) from the obvious one of a similar theory on the 14 dimensional exceptional compact Lie group G 2 .

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Section: Geometric Structures and Symmetry Breakingmentioning

confidence: 99%

Complex structure on the six dimensional sphere from a spontaneous symmetry breaking

Etesi

2015

Journal of Mathematical Physics

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“…It is important in the mathematical setting and is used in obvious settings when some class of Kähler or Hermitian manifolds is the central focus of investigation. The Gray-Hervella decomposition plays a role in the discussion of nearly Kähler and almost Kähler geometry as well as in the study of conformal equivalences among almost Hermitian structures (see for example [11,23], [4], and [5,7], respectively). It is related to the Tricerri-Vanhecke [28] decomposition of the curvature tensor in [12] and it has a prominent role in understanding the influence of the curvature on the underlying structure of the manifold [19].…”

Section: Introductionmentioning

confidence: 99%

“…The different classes have been considered for flag manifolds -they essentially reduce to four classes [26], and the 6-dimensional case has been considered in detail in [3]. The different classes of almost Hermitian structures also enter into the discussion of some harmonicity problems [5].…”

Section: Introductionmentioning

confidence: 99%

Geometric realizability of covariant derivative Kähler tensors for almost pseudo-Hermitian and almost para-Hermitian manifolds

Brozos‐Vázquez

Garcı́a-Rı́o

Gilkey

et al. 2011

Annali di Matematica

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“…The Hodge star operator defines an endomorphism * of Λ 2 R 4 with * 2 = Id. Hence we have the orthogonal decomposition (4) the transition matrix between these bases. Thanks to L. van Elfrikhof (1897), it is well-known that every matrix A in SO(4) can be represented as the product A = A 1 A 2 of two SO(4)-matrices of the following types…”

Section: The Twistor Space Of An Even-dimensional Riemannian Manifoldmentioning

confidence: 99%

“…([35]) The Calabi-Eckmann complex structure on S 2p+1 ×S 2q+1 with the product metric. Salvai[4] have given sufficient conditions for an almost Hermitian complex structure to minimize the energy functional among sections of the twistor bundle.Theorem 2. ([4]) Let (N, h) be a compact Riemannian manifold and let J be a compatible almost complex structure on it.…”

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confidence: 99%

Harmonic Almost Hermitian Structures

Davidov

2017

Springer INdAM Series

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This is a survey of old and new results on the problem when a compatible almost complex structure on a Riemannian manifold is a harmonic section or a harmonic map from the manifold into its twistor space. In this context, special attention is paid to the Atiyah-Hitchin-Singer and Eells-Salamon almost complex structures on the twistor space of an oriented Riemannian four-manifold. JOHANN DAVIDOVIn general, these critical points are not harmonic maps, but, by analogy, in [36] they are referred to as "harmonic almost complex structures". They are also called "harmonic sections" [35], a term, which is more appropriate in the context of this article.Forgetting the bundle structure of T, we can consider almost Hermitian structures that are critical points of the energy functional under variations through all maps N → T. These structures are genuine harmonic maps from (N, h) into (T, h 1 ) and we refer to [15] for basic fact about such maps.The main goal of this paper is to survey results about the harmonicity (in both senses) of the Atiyah-Hitchin-Singer and Eells-Salamon almost Hermitian structures on the twistor space of an oriented four-dimensional Riemannian manifold, as well as almost Hermitian structures on such a manifold.In Section 2 we recall some basic facts about the twistor spaces of even-dimensional Riemannian manifolds. Special attention is paid to the twistor spaces of oriented four-dimensional manifolds. In Sections 3 and 4 we discuss the energy functional on sections of a twistor space, i.e. almost Hermitian structures on the base Riemannian manifold. We state the Euler-Lagrange equation for such a structure to be a critical point of the energy functional (a harmonic section) obtained by Wood [35,36]. Several examples of non-Kähler almost Hermitian structures, which are harmonic sections are given. Kähler structures are absolute minima of the energy functional. G. Bor, L. Hernández-Lamoneda and M. Salvai [4] have given sufficient conditions for an almost Hermitian structure to be a minimizer of the energy functional. Their result (in fact, part of it) is presented in Section 4 and is used to supply examples of non-Kähler minimizers based on works by C. LeBrun [27] and I. Kim [24]. Section 4 ends with a lemma from [9], which rephrases the Euler-Lagrange equation for an almost Hermitian structure (h, J) on a manifold N in terms of the fundamental 2-form of (h, J) and the curvature of (N, h). This lemma has been used in [9] to show that the Atiyah-Hitchin-Singer almost Hermitian structure J 1 on the negative twistor space Z of an oriented Riemannian 4-manifold (M, g) is a harmonic section if and only if the base manifold (M, g) is self-dual, while the Eells-Salamon structure J 2 is a harmonic section if and only (M, g) is self-dual and of constant scalar curvature. The main part of the proof of this result (slightly different from the proof in [9]) is presented in Section 5. In this context, it is natural to ask when J 1 and J 2 are harmonic maps into the twistor space of Z. Recall that a map between Riem...

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Orthogonal almost-complex structures of minimal energy

Cited by 15 publications

References 15 publications

Complex structure on the six dimensional sphere from a spontaneous symmetry breaking

Complex structure on the six dimensional sphere from a spontaneous symmetry breaking

Geometric realizability of covariant derivative Kähler tensors for almost pseudo-Hermitian and almost para-Hermitian manifolds

Harmonic Almost Hermitian Structures

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