Unit vector fields on real space forms which are harmonic maps

Perrone, Domenico

doi:10.2140/pjm.2009.239.89

Cited by 10 publications

(5 citation statements)

References 26 publications

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“…Remark 2. The theorem generalizes the classical result that Hopf vector fields on odd dimensional spheres are harmonic maps (see [19] and also [22,1]). In fact, in the case (1, 2m), the cross product X is an orthogonal linear complex structure on R 2m and it may be thought of as a Hopf vector field on S 2m−1 : X (p) ∈ p ⊥ = T p S 2m−1 , identifying the Grassmannian G (1, 2m) with S 2m−1 and E 1 1,2m with T S 2m−1 .…”

Section: Introduction and Presentation Of The Resultssupporting

confidence: 74%

“…One of our two main theorems generalizes amply the classical result asserting that Hopf vector fields on odd dimensional spheres are harmonic maps into the unit tangent bundle endowed with the Sasaki metric [19] (see also [22,1]). Such a unit vector field is a critical point of the energy functional if one considers variations through all smooth mappings.…”

Section: Introduction and Presentation Of The Resultsmentioning

confidence: 63%

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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 Resultssupporting

confidence: 74%

Section: Introduction and Presentation Of The Resultsmentioning

confidence: 63%

Harmonic unit normal sections of Grassmannians associated with cross products

Ferraris¹,

Moas²,

Salvai³

2021

Preprint

View full text Add to dashboard Cite

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“…About the harmonicity of Hopf vector fields, Han and Yim [107] proved that these fields, namely, the unit Killing vector fields, are the unique unit vector fields on the unit sphere S 3 which define harmonic maps from S 3 to (T 1 S 3 ,ḡ 0 ), whereḡ 0 is the Sasaki metric. In [108], as a consequence of a more general result, we got in particular that Han-Yim's Theorem is invariant under a three-parameter deformation of the Sasaki metric on T 1 S 3 .…”

Section: Levi Harmonicity Of Reeb Vector Fieldsmentioning

confidence: 88%

Contact Semi-Riemannian Structures in CR Geometry: Some Aspects

Perrone

2019

Axioms

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There is one-to-one correspondence between contact semi-Riemannian structures ( η , ξ , φ , g ) and non-degenerate almost CR structures ( H , ϑ , J ) . In general, a non-degenerate almost CR structure is not a CR structure, that is, in general the integrability condition for H 1 , 0 : = X - i J X , X ∈ H is not satisfied. In this paper we give a survey on some known results, with the addition of some new results, on the geometry of contact semi-Riemannian manifolds, also in the context of the geometry of Levi non-degenerate almost CR manifolds of hypersurface type, emphasizing similarities and differences with respect to the Riemannian case.

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“…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%

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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Unit vector fields on real space forms which are harmonic maps

Cited by 10 publications

References 26 publications

Harmonic unit normal sections of Grassmannians associated with cross products

Harmonic unit normal sections of Grassmannians associated with cross products

Contact Semi-Riemannian Structures in CR Geometry: Some Aspects

Harmonic functions and quadratic harmonic morphisms on Walker spaces

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