Another Report on Harmonic Maps

Eells, James; Lemaire, Luc

doi:10.1112/blms/20.5.385

Cited by 437 publications

(100 citation statements)

References 5 publications

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“…Then the map f is said to be ± transversally holomorphic if df (J 1 (X)) = ±J 2 df (X), for all X ∈ TF ⊥ . Then as a simple application of the theorem of Lichnerowicz, (see [26,12, 32]), we obtain the following results. , 2), be two foliated transversally almost Hermitian manifolds, the foliation F 1 being minimal.…”

Section: Special Structuresmentioning

confidence: 95%

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Some Remarks on Transversally Harmonic Maps

Konderak¹,

Wolak²

2008

Glasgow Math. J.

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Abstract. We consider transversally harmonic foliated maps between two Riemannian manifolds equipped with Riemannian foliations. We give various characterisations of such maps and we study the relation between the properties "harmonic" and "transversally harmonic" for a given map. We also consider these problems for particular classes of manifolds: manifolds with transversally almost Hermitian foliations and Riemannian flows.2000 Mathematics Subject Classification. 53C12, 58E20.

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Section: Special Structuresmentioning

confidence: 95%

“…The equation trace∇df = 0 is the Euler-Lagrange equation of the variational problem defined by the energy functional. For details about harmonic maps see [13,10,11,12,1].…”

Section: Harmonic and Transversally Harmonic Mapsmentioning

confidence: 99%

Some Remarks on Transversally Harmonic Maps

Konderak¹,

Wolak²

2008

Glasgow Math. J.

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“…[6,4,5], that a map φ is harmonic if and only if its tension field is zero, i.e. if and only if a f iaa = 0.…”

Section: Definition 33 ([6]) a Smooth Map φ : M → N Is Said To Be Hmentioning

confidence: 99%

Restrictions on harmonic morphisms

Mustafa

1999

Conform. Geom. Dyn.

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Abstract. We consider horizontally (weakly) conformal maps φ between Riemannian manifolds and calculate a formula for the Laplacian of the dilation of φ, using the language of moving frames. Applying this formula to harmonic horizontally (weakly) conformal maps or equivalently to harmonic morphisms we obtain a Weitzenböck formula similar to an earlier result of the author (J. London Math. Soc. (2) 57 (1998), 746-756), and hence vanishing results for harmonic morphisms from compact manifolds of positive curvature. Further, a method is developed to obtain restrictions on harmonic morphisms from some non-compact and non-positively curved domains. Finally, a discussion of restrictions on harmonic morphisms between simply connected space forms is given.

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“…Thus our focus is not on regular harmonic maps and their many applications (see e.g. the surveys [34,35] and bibliography [13]) but rather on issues involving singularities.…”

Section: Introductionmentioning

confidence: 99%