Pseudo Harmonic Morphisms

Loubeau, E.

doi:10.1142/s0129167x97000457

Cited by 39 publications

(40 citation statements)

References 5 publications

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“…f -structures and isotropic quotients. The isotropy condition is effectively used in the theory of harmonic maps [9], [4], [1], [2]. The easiest case is the following:…”

Section: F -Structuresmentioning

confidence: 99%

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Holomorphic vector bundles on Kähler manifolds and totally geodesic foliations on Euclidean open domains

Aprodu

2015

Differential Geometry and its Applications

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Abstract. In this Note we establish a relation between sections in globally generated holomorphic vector bundles on Kähler manifolds, isotropic with respect to a nondegenerate quadratic form, and totally geodesic foliations on Euclidean open domains. We find a geometric condition for a totally geodesic foliation to originate in a holomorphic vector bundle. For codimension-two foliations, this description recovers [3] Theorem 2.8 and Theorem 2.15. The universal objects that play a key role are the orthogonal Grassmannians.In [3], P. Baird and J. C. Wood have solved completely a problem posed by Jacobi [8]. In modern language, the question was to classify harmonic morphisms with one-dimensional fibres from open domains in R 3 . In other words, it was asked to describe (locally) foliations in lines on open domains in R 3 which verify a geometric extracondition, horizontal weak conformality (HWC). Baird and Wood interpreted horizontal weak conformality as a holomorphic variation of the leaves. More precisely, assuming the foliation was simple, they associate a meromorphic map from the leaf space to a complex projective quadric and indicate an inverse construction. Two distinct cases are found, corresponding to whether or not the leaves pass through the origin.Their approach works for codimension-two foliations on higher-dimensional Euclidean open domains.In this Note, we extend the results of Baird and Wood to foliations of higher codimension. We establish a relation between holomorphic maps to an orthogonal Grassmannian, together with a section in the universal bundle, and certain foliations with totally geodesic leaves on Euclidean open domains, Theorem 3.1, Theorem 3.4. A key feature of the orthogonal Grassmannian is the universal property, it represents the functor of isomorphism classes of isotropic quotient bundles of trivial bundles, Proposition 1.6 in section 1. The definition and the basic properties of the orthogonal Grassmannians are recalled also in section 1.Notation and convention. All the manifolds are considered without boundary. If M is a real manifold, T R M denotes the real tangent bundle, and T C M := T R M ⊗ R C denotes the complexified tangent bundle. For a complex manifold N , we denote by T + N the holomorphic tangent bundle. Recall that, as real vector bundles, T R N and T + N are isomorphic, although they differ as subbundles in T C N . If ϕ : M → N is a submersion Date: February 24, 2018.

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“…f -structures and isotropic quotients. The isotropy condition is effectively used in the theory of harmonic maps [9], [4], [1], [2]. The easiest case is the following:…”

Section: F -Structuresmentioning

confidence: 99%

“…The latter condition is usually called in literature PHWC or pseudo-horizontally weakly conformal [9], [4], the reason being that it was introduced as an extension of horizontally weakly conformal maps.…”

Section: F -Structuresmentioning

confidence: 99%

Holomorphic vector bundles on Kähler manifolds and totally geodesic foliations on Euclidean open domains

Aprodu

2015

Differential Geometry and its Applications

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“…When N is endowed with an almost Hermitian structure J (so n is even), a class of mappings that includes the above ones was defined by [dϕ • dϕ t , J] = 0, cf. [27]. These maps are called pseudo horizontally weakly conformal maps (PHWC).…”

Section: Remark 22 It Is Easy To See Thatmentioning

confidence: 99%

On the geometrized Skyrme and Faddeev models

Slobodeanu

2010

Journal of Geometry and Physics

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Abstract. The higher-power derivative terms involved in both Faddeev and Skyrme energy functionals correspond to σ 2 -energy, introduced by Eells and Sampson in [13]. The paper provides a detailed study of the first and second variation formulae associated to this energy. Some classes of (stable) critical points are outlined. IntroductionCommon tools in field theory, non-linear σ-models are known in differential geometry mainly through the problem of harmonic maps between Riemannian manifolds. Namely a (smooth) mapping ϕ : (M, g) → (N, h) is harmonic if it is critical point for the Dirichlet energy functional [13],a generalization of the kinetic energy of classical mechanics. Less discussed from differential geometric point of view are Skyrme and Faddeev-Hopf models, which are σ-models with additional fourth-power derivative terms (for an overview including recent progress concerning both models, see [25]).The first one was proposed in the sixties by Tony Skyrme [37], to model baryons as topological solitons (see [31]) of pion fields. Meanwhile it has been shown [47] to be a low energy effective theory of quantum chromodynamics that becomes exact as the number of quark colours becomes large. Thus baryons are represented by energy minimising, topologically nontrivial maps ϕ :with the boundary condition ϕ({|x| → ∞}) = I 2 , called skyrmions. Their topological degree is identified with the baryon number. The static (conveniently renormalized) Skyrme energy functional isThis energy has a topological lower bound [14]: E Skyrme (ϕ) ≥ 6π 2 |degϕ|. In the second one, stated in 1975 by Ludvig Faddeev and Antti J. Niemi [15], the configuration fields are unitary vector fields ϕ : R 3 → S 2 ⊂ R 3 with the boundary condition ϕ({|x| → ∞}) = (0, 0, 1). The static energy in this case is given bywhere c 2 , c 4 are coupling constants. Again the field configurations are indexed by an integer, their Hopf invariant : Q(ϕ) ∈ π 3 (S 2 ) ∼ = Z and the energy has a topological lower bound: cf. [45]. Although this model can be viewed as a constrained variant of the Skyrme model, it exhibits important specific properties, e.g. it allows knotted solitons. Moreover, in [16] it has been proposed that it arises as a dual description of strongly coupled SU (2) Yang-Mills theory, with the solitonic strings (possibly) representing glueballs. See also [17] for an alternative approach to these issues.Both models rise the same kind of topologically constrained minimization problem: find out static energy minimizers in each topological class (i.e. of prescribed baryon number or Hopf invariant). We can give an unitary treatment for both if we take into account that they are particular cases of the following energy-type functional:

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“…A more systematic study of pseudo horizontally weakly conformal maps into almost Hermitian manifolds was carried out in [14] and [15] where the pull-back of the complex structure J is used to build an f -structure on the domain (M, g), i.e. an endomorphism F of the tangent bundle satisfying…”

Section: Introductionmentioning

confidence: 99%

Eigenvalues of harmonic almost submersions

Loubeau¹,

Slobodeanu

2009

Geom Dedicata

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Abstract. Maps between Riemannian manifolds which are submersions on a dense subset, are studied by means of the eigenvalues of the pull-back of the target metrics, the first fundamental form. Expressions for the derivatives of these eigenvalues yield characterizations of harmonicity, totally geodesic maps and biconformal changes of metric preserving harmonicity. A Schwarz lemma for pseudo harmonic morphisms is proved, using the dilatation of the eigenvalues and, in dimension five, a Bochner technique method, involving the Laplacian of the difference of the eigenvalues, gives conditions forcing pseudo harmonic morphisms to be harmonic morphisms.

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Pseudo Harmonic Morphisms

Cited by 39 publications

References 5 publications

Holomorphic vector bundles on Kähler manifolds and totally geodesic foliations on Euclidean open domains

Holomorphic vector bundles on Kähler manifolds and totally geodesic foliations on Euclidean open domains

On the geometrized Skyrme and Faddeev models

Eigenvalues of harmonic almost submersions

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