Harmonic morphisms between riemannian manifolds

Bent Fuglede

Annales de l'institut Fourier (1978)

  • Volume: 28, Issue: 2, page 107-144
  • ISSN: 0373-0956

Abstract

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A harmonic morphism f : M N between Riemannian manifolds M and N is by definition a continuous mappings which pulls back harmonic functions. It is assumed that dim M dim N , since otherwise every harmonic morphism is constant. It is shown that a harmonic morphism is the same as a harmonic mapping in the sense of Eells and Sampson with the further property of being semiconformal, that is, a conformal submersion of the points where d f vanishes. Every non-constant harmonic morphism is shown to be an open mapping.

How to cite

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Fuglede, Bent. "Harmonic morphisms between riemannian manifolds." Annales de l'institut Fourier 28.2 (1978): 107-144. <http://eudml.org/doc/74351>.

@article{Fuglede1978,
abstract = {A harmonic morphism $f : M\rightarrow N$ between Riemannian manifolds $M$ and $N$ is by definition a continuous mappings which pulls back harmonic functions. It is assumed that dim$M\ge $ dim$N$, since otherwise every harmonic morphism is constant. It is shown that a harmonic morphism is the same as a harmonic mapping in the sense of Eells and Sampson with the further property of being semiconformal, that is, a conformal submersion of the points where $df$ vanishes. Every non-constant harmonic morphism is shown to be an open mapping.},
author = {Fuglede, Bent},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {2},
pages = {107-144},
publisher = {Association des Annales de l'Institut Fourier},
title = {Harmonic morphisms between riemannian manifolds},
url = {http://eudml.org/doc/74351},
volume = {28},
year = {1978},
}

TY - JOUR
AU - Fuglede, Bent
TI - Harmonic morphisms between riemannian manifolds
JO - Annales de l'institut Fourier
PY - 1978
PB - Association des Annales de l'Institut Fourier
VL - 28
IS - 2
SP - 107
EP - 144
AB - A harmonic morphism $f : M\rightarrow N$ between Riemannian manifolds $M$ and $N$ is by definition a continuous mappings which pulls back harmonic functions. It is assumed that dim$M\ge $ dim$N$, since otherwise every harmonic morphism is constant. It is shown that a harmonic morphism is the same as a harmonic mapping in the sense of Eells and Sampson with the further property of being semiconformal, that is, a conformal submersion of the points where $df$ vanishes. Every non-constant harmonic morphism is shown to be an open mapping.
LA - eng
UR - http://eudml.org/doc/74351
ER -

References

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  8. [8] R. E. GREENE and H. WU, Embedding of open Riemannian manifolds by harmonic functions, Ann. Inst. Fourier, Grenoble, 25, 1 (1975), 215-235. Zbl0307.31003MR52 #3583
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  10. [10] O. D. KELLOGG, Foundations of potential theory, Berlin, Springer, 1929 (re-issued 1967). Zbl0152.31301JFM55.0282.01
  11. [11] J. LIOUVILLE, Note VI, p. 609-616 in G. Monge : Applications de l'Analyse à la Géométrie, 5e éd., Paris, 1850. 
  12. [12] Yu G. REŠETNJAK, O konformnyk otobrazenijah prostanstva. (Russian.) (On conformal mappings in space), Dokl. Akad. Nauk SSSR, 130 (1960), 1196-1198. (Sovjet Math., 1 (1960), 153-155.) Zbl0099.37901
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Citations in EuDML Documents

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  1. Laurent Danielo, Construction de métriques d’Einstein à partir de transformations biconformes
  2. A. Mohammed Cherif, Djaa Mustapha, On generalized f -harmonic morphisms
  3. Jean-Marie Burel, Almost symplectic structures and harmonic morphisms
  4. R. W. R. Darling, Martingales in manifolds. Definition, examples and behaviour under maps
  5. Paul Baird, Harmonic morphisms onto Riemann surfaces and generalized analytic functions
  6. Bernt Oksendal, L. Csink, Stochastic harmonic morphisms : functions mapping the paths of one diffusion into the paths of another
  7. Frédérique Duheille, Une preuve probabiliste élémentaire d'un résultat de P. Baird et J. C. Wood
  8. Mustapha Chadli, Mohamed El Kadiri, Sabah Haddad, Biharmonic morphisms
  9. Ye-Lin Ou, Quadratic harmonic morphisms and O-systems
  10. Paul Baird, Harmonic morphisms and circle actions on 3- and 4-manifolds

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