Harmonicity and minimality of oriented distributions

Gil-Medrano, Olga; González–Dávila, J. C.; Vanhecke, Lieven

doi:10.1007/bf02803502

Cited by 20 publications

(19 citation statements)

References 21 publications

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“…Then σ is spanned by vector X. One can show that σ is harmonic with respect to standard inner product g on S 3 [4,3]. Suppose there exists function µ such that σ is harmonic with respect to e 2µ g, where µ is not a constant.…”

Section: Proof Follows Immediately By (8)mentioning

confidence: 99%

“…(1) generalized Hopf fibrations [3,4], (2) characteristic distribution of a contact structure Notice that example (2) is a special case of example (4) since Reeb vector field of a contact structure is unit harmonic.…”

Section: Introductionmentioning

confidence: 99%

“…(1) generalized Hopf fibrations [3,4], (2) characteristic distribution of a contact structure [4], (3) invariant distributions on compact semisimple Lie groups [2]. (4) distributions orthogonal to harmonic unit vector fields [4].…”

Section: Introductionmentioning

confidence: 99%

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Two notes on harmonic distributions

Niedziałomski

2014

Differential Geometry and its Applications

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Abstract. We say that a distribution is harmonic if it is harmonic when considered as a section of a Grassmann bundle. We find new examples of harmonic distributions and show nonexistense of harmonic distrubutions on some Riemannian manifolds by two different approaches. Firstly, we lift distributions to the second tangent bundle equipped with the Sasaki metric. Secondly, we deform conformally the metric on a base manifold. IntroductionHarmonic map σ : M → N between Riemannian manifolds is a critical point of energy functionalWhen the considered map σ is a section of a submersion, we may define the weaker condition of harmonicity. In the tangent space of any submersion we may distinguish the vertical subspace and therefore we may consider the vertical projection σ V * (X) of a vector σ * (X), X ∈ T x M. Thus, we define harmonicity of a section via vanishing of the Euler-Lagrange equation of the vertical energy functionalHarmonic distribution (plane field) σ on a Riemannian manifold M is a distribution, which considered as a section σ : M → Gr p (M), p = dim σ, of a Grassmann bundle is harmonic (We equip Gr p (M) with the Riemannian metric induced by the Riemannian metric on M and the invariant inner product on O(n)). The following distributions are known to be harmonic:(1) generalized Hopf fibrations [3,4], (2) characteristic distribution of a contact structure Notice that example (2) is a special case of example (4) since Reeb vector field of a contact structure is unit harmonic.2000 Mathematics Subject Classification. 53C43, 58E20.

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Section: Proof Follows Immediately By (8)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Two notes on harmonic distributions

Niedziałomski

2014

Differential Geometry and its Applications

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“…Hence, using Lemma 2.1, we have (see also [14]) Proposition 3.2 Let π : E → M be a vector bundle with a metric connection over a closed and oriented Riemannian manifold and σ ∈ Γ ∞ (S E (r )). Then, we have:…”

Section: Harmonicity Of Sections Of Sphere Bundlesmentioning

confidence: 99%

“…In [14], Gil-Medrano et al considered the energy functional defined on (r, s)-tensorial bundles on M which is a particular case of the vector bundles considered here. The characterising condition for critical points of the energy functional on the space of sections of sphere bundles was shown in [28].…”

Section: Introductionmentioning

confidence: 99%

Harmonicity of sections of sphere bundles

González–Dávila

Cabrera

Salvai³

2008

Math. Z.

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We consider the energy functional on the space of sections of a sphere bundle over a Riemannian manifold (M, ·, · ) equipped with the Sasaki metric and discuss the characterising condition for critical points. Furthermore, we provide a useful method for computing the tension field in some particular situations. Such a method is shown to be adequate for many tensor fields defined on manifolds M equipped with a G-structure compatible with ·, · . This leads to the construction of several new examples of differential forms which are harmonic sections or determine a harmonic map from (M, ·, · ) into its sphere bundle.

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On the frame bundle adapted to a submanifold

Niedziałomski

2014

Math. Nachr.

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Let M be a submanifold of a Riemannian manifold (N,g). M induces a subbundle O(M,N) of adapted frames over M of the bundle of orthonormal frames O(N). The Riemannian metric g induces a natural metric on O(N). We study the geometry of a submanifold O(M,N) in O(N). We characterize the horizontal distribution of O(M,N) and state its correspondence with the horizontal lift in O(N) induced by the Levi–Civita connection on N. In the case of extrinsic geometry, we show that minimality is equivalent to harmonicity of the Gauss map of the submanifold M with a deformed Riemannian metric. In the case of intrinsic geometry we compute the curvatures and compare this geometry with the geometry of M.

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Harmonicity and minimality of oriented distributions

Cited by 20 publications

References 21 publications

Two notes on harmonic distributions

Two notes on harmonic distributions

Harmonicity of sections of sphere bundles

On the frame bundle adapted to a submanifold

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