Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-24T13:35:56.446Z Has data issue: false hasContentIssue false
  • English
  • Français

On the geometry of harmonic morphisms

Published online by Cambridge University Press:  24 October 2008

Sigmundur Gudmundsson
Sigmundur Gudmundsson
Affiliation:
Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT
Get access Rights & Permissions [Opens in a new window]

Abstract

Let π:M→B be a horizontally conformal submersion. We give necessary curvature conditions on the manifolds M and B, which lead to non-existence results for certain horizontally conformal maps, and harmonic morphisms. We then classify all such maps between open subsets of Euclidean spaces, which additionally have totally geodesic fibres and are horizontally homothetic. They are orthogonal projections on each connected component, followed by a homothety.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 108 , Issue 3 , November 1990 , pp. 461 - 466
Copyright
Copyright © Cambridge Philosophical Society 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Baird, P.. Harmonic Maps with Symmetry, Harmonic Morphisms and Deformations of Metrics. Research Notes in Math. no. 87 (Pitman, 1983).Google Scholar
[2]Baird, P. and Eells, J.. A conservation law for harmonic maps. In Geometry Symposium Utrecht 1980, Lecture Notes in Math. vol. 894 (Springer-Verlag, 1981), pp. 1–25.CrossRefGoogle Scholar
[3]Baird, P. and Wood, J. C.. Bernstein theorems for harmonic morphisms from ℕ3 and S3. Math. Ann. 280 (1988), 579–603.CrossRefGoogle Scholar
[4]Baird, P. and Wood, J. C.. Harmonic morphisms and conformal foliation by geodesies of three-dimensional space forms. J. Austral. Math. Soc. (To appear.)Google Scholar
[5]Baird, P. and Wood, J. C.. Harmonic morphisms, Seifert fibre spaces and conformal foliations. (Preprint, University of Leeds 1990.)Google Scholar
[6]Besse, A.. Einstein Manifolds (Springer-Verlag, 1987).CrossRefGoogle Scholar
[7]Bivens, I.. Orthogonal geodesic and minimal distributions. Trans. Amer. Math. Soc. 275 (1983), 397–408.CrossRefGoogle Scholar
[8]Eells, J. and Lemaire, L.. Another report on harmonic maps. Bull. London Math. Soc. 20 (1988), 385–524.CrossRefGoogle Scholar
[9]Fuglede, B.. Harmonic morphisms between Riemannian manifolds. Ann. Inst. Fourier (Grenoble) 28 (1978), 107–144.CrossRefGoogle Scholar
[10]Fuglede, B.. A criterion of non-vanishing differential of a smooth map. Bull. London Math. Soc. 14 (1982), 98–102.CrossRefGoogle Scholar
[11]Gromoll, D., Klingenberg, W. and Meyer, W.. Riemannsche Geometrie im Großen. Lecture Notes in Math. vol. 55 (Springer-Verlag, 1975).CrossRefGoogle Scholar
[12]Gudmundsson, S.. Ph.D. thesis, University of Leeds (in preparation).Google Scholar
[13]Ishihara, T.. A mapping of Riemannian manifolds which preserves harmonic functions. J. Math. Kyoto Univ. 19 (1979), 215–229.Google Scholar
[14]Johnson, D. L. and Whitt, L. B.. Totally geodesic foliations. J. Differential Geom. 15 (1980), 225–235.Google Scholar
[15]Kasue, A. and Washio, T.. Growth of equivariant harmonic maps and harmonic morphisms. (Preprint.)Google Scholar
[16]O'Neill, B.. The fundamental equations of a submersion. Michigan Math. J. 13 (1966), 459–469.CrossRefGoogle Scholar