Some Remarks on Transversally Harmonic Maps

Konderak, Jerzy J.; Wolak, Robert

doi:10.1017/s001708950700403x

Cited by 21 publications

(14 citation statements)

References 35 publications

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“…Taking into account our previous considerations we can formulate the following theorems, whose proofs can be found in [18,19]:…”

Section: Transversely Harmonic Mapsmentioning

confidence: 99%

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Orbifolds, geometric structures and foliations. Applications to harmonic maps

Wolak

2016

Preprint

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In recent years a lot of attention has been paid to topological spaces which are a bit more general than smooth manifolds -orbifolds. Orbifolds are intuitively speaking manifolds with some singularities. The formal definition is also modelled on that of manifolds, an orbifold is a topological space which locally is homeomorphic to the orbit space of a finite group acting on R n . Orbifolds were defined by Satake, as V-manifolds, cf. [25], then studied by W. Thurston, cf. [29], who introduced the term "orbifold". Due to their importance in physics, and in particular in the string theory, orbifolds have been drawing more and more attention. In this paper we propose to show that the classical theory of geometrical structures, cf. [27,15,22], easily translates itself to the context of orbifolds and is closely related to the theory of foliated geometrical structures, cf. [32]. Finally, we propose a foliated approach to the study of harmonic maps between Riemannian orbifolds based on our previous research into transversely harmonic maps, cf. [18,19].

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“…Taking into account our previous considerations we can formulate the following theorems, whose proofs can be found in [18,19]:…”

Section: Transversely Harmonic Mapsmentioning

confidence: 99%

“…Finally, we propose a foliated approach to the study of harmonic maps between Riemannian orbifolds based on our previous research into transversely harmonic maps, cf. [18,19].…”

mentioning

confidence: 99%

Orbifolds, geometric structures and foliations. Applications to harmonic maps

Wolak

2016

Preprint

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“…The relationship between harmonicity and transverse harmonicity has been studied by Konderak and Wolak [13] based on their definition of transverse harmonicity. Since our definition contains the mean curvature κ additionally, we can get the similar relationship which is more concise.…”

Section: Proposition 21 (Cf [2]) the Euler-lagrange Equation For Tmentioning

confidence: 99%

Bochner-Type Formulas for Transversally Harmonic Maps

Chen

Zhou

2012

Int. J. Math.

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The main purpose of this paper is to study the properties of transversally harmonic maps by using Bochner-type formulas. As an application, we obtain the following theorem between compact Sasaki manifolds: Let f be a transversally harmonic map from compact Sasaki manifold M to compact Sasaki manifold M , and M has a strongly negative transverse curvature. If the rank of d T f is at least three at some points of M , then f is contact holomorphic (or contact anti-holomorphic).

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“…Then φ is said to be transversally harmonic if the transversal tension field τ b (φ) = tr Q (∇ tr d T φ) vanishes, where d T φ = dφ| Q and Q is the normal bundle of F . Transversally harmonic maps on foliated Riemannian manifolds have been studied by many authors [3,12,13,18]. However, a transversally harmonic map is not a critical point of the transversal energy [10]…”

Section: Introductionmentioning

confidence: 99%

Liouville type theorem for (F;F')p-harmonic maps on foliations

Fu¹,

Jung²

2022

Preprint

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In this paper, we study (F , F ′ ) p -harmonic maps between foliated Riemannian manifolds (M, g, F ) andis a critical point of the transversal p-energy E B,p (φ), which is a generalization of (F , F ′ )harmonic map. Precisely, we give the first and second variational formulas for (F , F ′ ) pharmonic maps. We also investigate the generalized Weitzenböck type formula and the Liouville type theorem for (F , F ′ ) p -harmonic map. E B,p (φ) = 1 p ˆM |d T φ| p µ M .

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Some Remarks on Transversally Harmonic Maps

Cited by 21 publications

References 35 publications

Orbifolds, geometric structures and foliations. Applications to harmonic maps

Orbifolds, geometric structures and foliations. Applications to harmonic maps

Bochner-Type Formulas for Transversally Harmonic Maps

Liouville type theorem for (F;F')p-harmonic maps on foliations

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