Harmonicity and minimality of distributions on Riemannian manifolds via the intrinsic torsion

González–Dávila, J. C.

doi:10.4171/rmi/777

Cited by 11 publications

(11 citation statements)

References 22 publications

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“…sense were made for a wide variety of structures (mainly for unit vector fields, but also, for example, for distributions, metric contact structures, etc. ), see for instance [4,7,9,10,11,14,21].…”

Section: Introduction and Presentation Of The Resultsmentioning

confidence: 99%

Harmonic unit normal sections of Grassmannians associated with cross products

Ferraris¹,

Moas²,

Salvai³

2021

Preprint

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Let G (k, n) be the Grassmannian of oriented subspaces of dimension k of R n with its canonical Riemannian metric. We study the energy of maps assigning to each P ∈ G (k, n) a unit vector normal to P . They are sections of a sphere bundle E 1 k,n over G (k, n). The octonionic double and triple cross products induce in a natural way such sections for k = 2, n = 7 and k = 3, n = 8, respectively. We prove that they are harmonic maps into E 1 k,n endowed with the Sasaki metric. This, together with the well-known result that Hopf vector fields on odd dimensional spheres are harmonic maps into their unit tangent bundles, allows us to conclude that all unit normal sections of the Grassmannians associated with cross products are harmonic. In a second instance we analyze the energy of maps assigning an orthogonal complex structure J (P ) on P ⊥ to each P ∈ G (2, 8). We prove that the one induced by the octonionic triple product is a harmonic map into a suitable sphere bundle over G (2, 8). This generalizes the harmonicity of the canonical almost complex structure of S 6 .

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Section: Introduction and Presentation Of The Resultsmentioning

confidence: 99%

Harmonic unit normal sections of Grassmannians associated with cross products

Ferraris¹,

Moas²,

Salvai³

2021

Preprint

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“…Proof. Using that U(ξ, ξ) = 0, D determines a Riemannian foliation (see [9,Lemma 4.1]) and ξ must be geodesic. If D were totally geodesic, we would equivalently have U(D, D) ⊂ D, but this condition would imply that η c (ξ) = 0, which is a contradiction.…”

Section: Canonical Foliations Of Homogeneous Riemannian Manifoldsmentioning

confidence: 99%

Cyclic homogeneous Riemannian manifolds

Gadea

González–Dávila

Oubiña

2015

Annali di Matematica

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Abstract. In spin geometry, traceless cyclic homogeneous Riemannian manifolds equipped with a homogeneous spin structure can be viewed as the simplest manifolds after Riemannian symmetric spin spaces. In this paper, we give some characterizations and properties of cyclic and traceless cyclic homogeneous Riemannian manifolds and we obtain the classification of simplyconnected cyclic homogeneous Riemannian manifolds of dimension less than or equal to four. We also present a wide list of examples of non-compact irreducible Riemannian 3-symmetric spaces admitting cyclic metrics and give the expression of these metrics.

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“…We say that π is a transversally symmetric fibration [8] if (G, L) is a Riemannian symmetric pair. Moreover, π is said to be of compact type, noncompact type or Euclidean type according to the type of (G, L).…”

Section: Introductionmentioning

confidence: 99%

Homogeneous spin Riemannian manifolds with the simplest Dirac operator

Gadea

González–Dávila

Oubiña

2018

Advances in Geometry

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We show the existence of nonsymmetric homogeneous spin Riemannian manifolds whose Dirac operator is like that on a Riemannian symmetric spin space. Such manifolds are exactly the homogeneous spin Riemannian manifolds (M, g) which are traceless cyclic with respect to some quotient expression M = G/K and reductive decomposition g = k⊕m. Using transversally symmetric fibrations of noncompact type, we give a list of them.Keywords: Dirac operator, homogeneous spin Riemannian manifolds, traceless cyclic homogeneous Riemannian manifolds, Riemannian symmetric spaces.MSC 2010: 53C30, 53C35, 34L40.Acknowledgments: The first and second authors have been supported by DGI (Spain) Project MTM2013-46961-P.

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Harmonicity and minimality of distributions on Riemannian manifolds via the intrinsic torsion

Cited by 11 publications

References 22 publications

Harmonic unit normal sections of Grassmannians associated with cross products

Harmonic unit normal sections of Grassmannians associated with cross products

Cyclic homogeneous Riemannian manifolds

Homogeneous spin Riemannian manifolds with the simplest Dirac operator

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