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Mean Curvature of Riemannian Foliations

Published online by Cambridge University Press:  20 November 2018

Peter March ,
Maung Min-Oo  and
Ernst A. Ruh
Peter March
Affiliation:
Department of Mathematics, The Ohio State University, Columbus OH 43210, U.S.A.
Maung Min-Oo
Affiliation:
Math. Inst., Univ. Fribourg, Chemin du Musee 23, CH-1700 Fribourg, Switzerland
Ernst A. Ruh
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1
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Abstract

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It is shown that a suitable conformai change of the metric in the leaf direction of a transversally oriented Riemannian foliation on a closed manifold will make the basic component of the mean curvature harmonic. As a corollary, we deduce vanishing and finiteness theorems for Riemannian foliations without assuming the harmonicity of the basic mean curvature.

Keywords

53C1257R30
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 39 , Issue 1 , 01 March 1996 , pp. 95 - 105
Copyright
Copyright © Canadian Mathematical Society 1995

References

[A] Alvarez López, J. A., The basic component of the mean curvature of Riemannian foliations, Ann. Global Analysis and Geom. 10(1992), 179–194.Google Scholar
[AT] Alvarez-López, J. A. and Tondeur, P., Hodge decomposition along the leaves of a Riemannian foliation, J. Funct. Anal. 99(1991), 443–458.Google Scholar
[KT1] Kamber, F. W. and Tondeur, Ph., Foliations and metrics, Proceedings, Univ. of Maryland, Birkhàuser, Progress in Math. 32(1983), 103–152.Google Scholar
[KT2] Kamber, F. W., Duality for foliations, Astérisque 116(1984), 108–116.Google Scholar
[KT3] Kamber, F. W., DeRham-Hodge theory for Riemannian foliations, Indiana Univ. Math. J. 35(1986), 321–331.Google Scholar
[KR] Krein, M. G. and Rutman, M. A., Linear Operators leaving invariant a cone in Banach space, A.M.S. Translations Ser. 1, Vol. 10(1962), 199–325.Google Scholar
[MMR] March, P., Min-Oo, M. and Ruh, E. A., Mean curvature of Riemannian foliations, Ohio State Math. Res. Inst. Preprint No. 8(1991).Google Scholar
[MRT] Min-Oo, M., Ruh, E. A. and Tondeur, P., Vanishing theorems for the basic cohomology of Riemannian foliations, J. Reine Angew. Math. 415(1991), 167–174.Google Scholar
[M] Molino, P., Riemannian Foliations, Birkhàuser, Boston, 1988.Google Scholar
[Ma] Masa, X., Duality and Minimality in Riemannian foliations, Comment. Math. Helv. 67(1992), 17—27.Google Scholar
[N] Neveu, J., Mathematical Foundations of the Calculus of Probability, Holden Day, San Fransisco, etc., 1965.Google Scholar
[R] Reihart, B., Differential geometry of foliations, Ergeb. Math. 99, Springer Verlag, Berlin, etc., 1983.Google Scholar
[Ru] Rummler, H.-K., Quelques notions simples en géométrie at leur applications aux feuilletages compacts, Comment. Math. Helv. 54(1979), 224–238.Google Scholar
[T] Tondeur, P., Foliations on Riemannian manifolds, Universitext, Springer Verlag, Berlin, etc., 1988.Google Scholar