Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-01-07T16:05:20.575Z Has data issue: false hasContentIssue false
  • English
  • Français

Harmonic morphisms between spaces of constant curvature

Published online by Cambridge University Press:  20 January 2009

Sigmundur Gudmundsson
Sigmundur Gudmundsson
Affiliation:
Department of Pure MathematicsUniversity of LeedsLeeds LS2 9JT, England Department of MathematicsScience InsituteUniversity of IcelandDunhaga 3107 Reykjavik, Iceland
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let M and N be simply connected space forms, and U an open and connected subset of M. Further let π: U → N be a horizontally homothetic harmonic morphism. In this paper we show that if π has totally geodesic fibres and integrable horizontal distribution, then the horizontal foliation of U is totally umbilic and isoparametric. This leads to a classification of such maps. We also show that horizontally homothetic harmonic morphisms of codimension one are either Riemannian submersions modulo a constant, or up to isometrics of M and N one of six well known examples.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 36 , Issue 1 , February 1993 , pp. 133 - 143
Copyright
Copyright © Edinburgh Mathematical Society 1993

References

REFERENCES

1.Baird, P. and Eells, J., A conservation law for harmonic maps, in Geometry Symposium Utrecht 1980 (Lecture Notes in Math. 894, Springer-Verlag, 1981) 1–25.Google Scholar
2.Baird, P. and Wood, J. C., Bernstein theorems for harmonic morphisms from R3 and S3, Math. Ann. 280 (1988), 579–603.CrossRefGoogle Scholar
3.Baird, P. and Wood, J. C., Harmonic morphisms and conformal foliation by geodesies of three-dimensional space forms, J. Australian Math. Soc. Ser. A 51 (1991), 118–153.CrossRefGoogle Scholar
4.Baird, P. and Wood, J. C., Harmonic morphisms, Siefert spaces and conformal foliations, Proc. London Math. Soc. 64 (1992), 170–196.Google Scholar
5.Baird, P. and Wood, J. C., The geometry of a pair of Riemannian foliations by geodesies and associated harmonic morphisms, Bull. Soc. Math. Belg., to appear.Google Scholar
6.Bernard, A., Campbell, E. A. and Davie, A. M., Brownian motion and generalized analytic and inner functions, Ann. Inst. Fourier (Grenoble) 29 (1979), 207–228.Google Scholar
7.Cartan, E., Families de surfaces isoparamétriques dans les espaces á courbure constante, Annali di Mat. 17 (1938), 177–191.Google Scholar
8.Cartan, E., Sur des families remarquables d'hypersurfaces isoparamétrique dans les espaces sphérique, Math. Z. 45 (1939), 335–367.CrossRefGoogle Scholar
9.Carter, S. and West, A., Generalized Cartan polynomials, J. London Math. Soc. (2) 32 (1984), 305–316.Google Scholar
10.Carter, S. and West, A., Isoparametric systems and transnormality Proc. Land. Math. Soc. (3) 51 (1985), 520–542.CrossRefGoogle Scholar
11.Darling, R. W. R., Martingales in manifolds—Definition, examples and behaviour under maps, in Seminaire de Probabilites XVI 1980/81. Supplement: Geometrie Differentielle Stochastique (Springer Lecture Notes in Mathematics 921, 1982).Google Scholar
12.Eells, J. and Lemaire, L., Another report on harmonic maps, Bul. London Math. Soc. 20 (1988), 385–524.Google Scholar
13.Escobales, R. H., Riemannian submersions with totally geodesic fibres, J. Differential Geom. 10 (1975), 253–276.CrossRefGoogle Scholar
14.Fuglede, B., Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier (Grenoble) 28 (1978), 107–144.CrossRefGoogle Scholar
15.Fuglede, B., A criterion of non-vanishing differential of a smooth map, Bull. London Math. Soc. 32 (1982), 98–102.Google Scholar
16.Gudmundsson, S., On the geometry of harmonic morphisms, Math. Proc. Cambridge Philos Soc. 108 (1990), 461–466.Google Scholar
17.Gudmundsson, S., The Geometry of Harmonic Morphisms, Ph.D. thesis, University of Leeds 1992.Google Scholar
18.Ishihara, T., A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ. 19 (1979), 215–229.Google Scholar
19.Kasue, A. and Washio, T., Growth of equivariant harmonic maps and harmonic morphisms, Osaka J. Math. 27 (1990), 899–928.Google Scholar
20.Lévy, P., Processus Stochastique et Mouvement Brownien (Gauthier-Villars, Paris, 1948).Google Scholar
21.O'Neill, B., The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459–469.CrossRefGoogle Scholar
22.Palais, R. S. and Terng, C.-I., Critical Point Theory and Submanifolds Geometry (Springer Lecture Notes in Mathematics 1353, Springer-Verlag, 1988).CrossRefGoogle Scholar
23.Spivak, M., A Comprehensive Introduction to Differential Geometry IV (Publish or Perish, 1979).Google Scholar
24.Terng, C.-I., Isoparametric submanifolds and the Coxeter groups, J. Differential Geom. 21 (1985), 79–107.CrossRefGoogle Scholar
25.Wood, J. C., Harmonic morphisms, foliations and Gauss maps, Contemp. Math. 49 (1986), 145–184.Google Scholar