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Harmonic morphisms and conformal foliations by geodesics of three-dimensional space forms

Part of: Variational problems in infinite-dimensional spaces Global differential geometry Higher-dimensional theory

Published online by Cambridge University Press:  09 April 2009

Paul Baird  and
John C. Wood
John C. Wood
Affiliation:
Department of Pure Mathematics University of LeedsLeeds LS29JT United Kingdom
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Abstract

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A complete classification is given of harmonic morphisms to a surface and conformal foliations by geodesics, with or without isolated singularities, of a simply-connected space form. The method is to associate to any such a holomorphic map from a Riemann surface into the space of geodesics of the space form. Properties such as nonintersecting fibres (or leaves) are translated into conditions on the holomorphic mapping which show it must have a simple form corresponding to a standard example.

MSC classification

Secondary: 53C12: Foliations (differential geometric aspects) 58E20: Harmonic maps , etc. 31B99: None of the above, but in this section

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 51 , Issue 1 , August 1991 , pp. 118 - 153
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Baird, P., ‘Harmonic morphisms onto Riemann surfaces and generalized analytic functions’, Ann. Inst. Fourier (Grenoble) 37 (1987), 135–173.CrossRefGoogle Scholar
[2]Baird, P., ‘Harmonic morphisms and circle actions on 3- and 4-manifolds’, Ann. Inst. Fourier (Grenoble) 40 (1990), 177–212.Google Scholar
[3]Baird, P. and Eells, J., A conservation law for harmonic maps, in: Looijenga, E., Siersma, D., Takens, F. (eds.), Geometry Symposium Utrecht 1980, Proceedings, Lecture Notes in Mathematics, Vol. 894, pp. 1–25, Springer, Berlin, Heidelberg, New York, 1981.Google Scholar
[4]Baird, P. and Wood, J. C., ‘Bernstein theorems for harmonic morphisms from R 3 and S 3’, Math. Ann. 280 (1988), 579–603.CrossRefGoogle Scholar
[5]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
[6]Conway, J. B., Functions of one complex variable, Springer, Berlin, Heidelberg, New York, 1983.Google Scholar
[7]Eells, J. and Polking, J. C., ‘Removable singularities of harmonic maps’, Indiana Univ. Math. J. 33 (1984), pp. 859–871.Google Scholar
[8]Eells, J. and Sampson, J. H., ‘Harmonic mappings of Riemannian manifolds,’ Amer. J. Math. 86 (1964), 109–160.Google Scholar
[9]Fuglede, B., ‘Harmonic morphisms between Riemannian manifolds’, Ann. Inst. Fourier (Grenoble) 28 (1978), 107–144.Google Scholar
[10]Greenberg, M., Lectures on algebraic topology, Benjamin, New York, Amsterdam, 1966.Google Scholar
[11]Greene, R. E. and Wu, H., ‘Embeddings of open Riemannian manifolds by harmonic functions’, Ann. Inst. Fourier (Grenoble) 12 (1962), 415–471.Google Scholar
[12]Hector, G. and Hirsch, U., Introduction to the geometry of foliations, parts A and B, Aspects of Mathematics, Vieweg, Brauschweig, Wiesbaden, 1981.CrossRefGoogle Scholar
[13]Heins, M., Complex function theory, Academic Press, 1968.Google Scholar
[14]Helms, L. L., Introduction to potential theory, Wiley, New York, London, Sydney, Toronto, 1969.Google Scholar
[15]Hitchin, N., ‘Monopoles and geodesics’, Comm. Math. Phys. 83 (1982), 579–602.Google Scholar
[16]Ishihara, T., ‘A mapping of Riemannian manifolds which preserves harmonic functions’, J. Math. Kyoto University 19 (1979), 215–229.Google Scholar
[17]Morgan, A., ‘Holonomy and metric properties of foliations in higher codimensions’, Proc. Amer. Math. Soc. 58 (1976), 255–261.CrossRefGoogle Scholar
[18]Rienhart, B. L., Differential geometry of foliations, Ergebnisse Math., Vol. 99, Springer-Verlag, Berlin, Heidelberg, New York, 1983.Google Scholar
[19]Steenrod, N., The topology of fibre bundles, Princeton University Press, 1951.Google Scholar
[20]Vaisman, I., ‘Conformal foliations’, Kodai Math. J. 2 (1979), 26–37.Google Scholar
[21]Wolf, J., Spaces of constant curvature, McGraw-Hill, New York, St. Louis, San Francisco, Toronto, London, Sydney, 1967.Google Scholar
[22]Wood, J. C., Harmonic morphisms foliations and Gauss maps, in: Siu, Y. T. (ed.), Complex differential geometry and nonlinear differential equations, Contemporary Mathematics, vol. 49, pp. 145–183, Amer. Math. Soc., Providence, R.I., 1986.Google Scholar