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Harmonicity of a foliation and of an associated map

Published online by Cambridge University Press:  17 April 2009

Philippe Tondeur  and
Lieven Vanhecke
Philippe Tondeur
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, II 61801, United States of America
Lieven Vanhecke
Affiliation:
Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium
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Abstract

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A foliation on a Riemannian manifold (M, g) is harmonic if all the leaves are minimal submanifolds. We give a new characterisation of the harmonicity of a foliation on (M, g) by the harmonicity of an associated bundle map of (T M, gC), where gC is the complete lift metric of g to the tangent bundle as introduced by Yano and Ishihara.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 54 , Issue 2 , October 1996 , pp. 241 - 246
Copyright
Copyright © Australian Mathematical Society 1996

References

REFERENCES

[1]Besse, A.L., Einstein manifolds, Ergeb. Math. Grenzgeb. 3. Folge 10 (Springer-Verlag, Berlin, Heidelberg, New York, 1987).CrossRefGoogle Scholar
[2]Eells, J. and Lemaire, L., ‘A report on harmonic maps’, Bull. London Math. Soc. 10 (1978), 1–68.CrossRefGoogle Scholar
[3]Eells, J. and Lemaire, L., ‘Another report on harmonic maps’, Bull. London Math. Soc. 20 (1988), 385–524.CrossRefGoogle Scholar
[4]Eells, J. and Sampson, J.H., ‘Harmonic mappings of Riemannian manifolds’, Amer. J. Math. 86 (1964), 109–160.CrossRefGoogle Scholar
[5]García-Río, E., Vanhecke, L. and Vázquez-Abal, M.E., ‘Harmonic endomorphism fields’, Illinois J. Math. (to appear).Google Scholar
[6]Gray, A., ‘Pseudo-Riemannian almost product manifolds and submersions’, J. Math. Mech. 16 (1967), 715–737.Google Scholar
[7]Kamber, F. and Tondeur, Ph., ‘Harmonic foliations’, Lecture Notes in Math. 949 (1982), 87–121.CrossRefGoogle Scholar
[8]Kamber, F. and Tondeur, Ph., ‘Foliations and metrics’, Progr. Math. 32 (1983), 103–152.Google Scholar
[9]Masa, X., ‘Duality and minimality in Riemannian foliations’, Comment. Math. Helv. 67 (1992), 17–27.CrossRefGoogle Scholar
[10]O'Neill, B., ‘The fundamental equations of a submersion’, Michigan Math. J. 13 (1966) 459–469.CrossRefGoogle Scholar
[11]Reinhart, B.L., Differential geometry of foliations, Ergeb. Math. Grenzgeb. 99 (Springer-Verlag, Berlin, Heidelberg, New York, 1983).CrossRefGoogle Scholar
[12]Rummler, R., ‘Quelques notions simples en géométrie riemannienne et leurs applications aux feuilletages compacts’, Comment. Math. Helv. 54 (1979), 224–239.CrossRefGoogle Scholar
[13]Sullivan, D., ‘A homological characterization of foliations consisting of minimal surfaces’, Comment. Math. Helv. 54 (1979), 218–223.CrossRefGoogle Scholar
[14]Tondeur, Ph., Foliations on Riemannian manifolds, Universitext (Springer-Verlag, Berlin, Heidelberg, New York, 1988).CrossRefGoogle Scholar
[15]Tondeur, Ph., ‘Geometry of Riemannian foliations’, in Seminar on Math. Sciences 20 (Keio University, Yokohama, 1994).Google Scholar
[16]Tondeur, Ph., ‘Riemannian foliations and tautness’, Proc. Sympos. Pure Math. 54 (1993), 667–672.CrossRefGoogle Scholar
[17]Yano, K. and Ishihara, S., Tangent and cotangent bundles, Pure and Appl. Math. 16 (Marcel Dekker, New York, 1973).Google Scholar