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Further geometry of the mean curvature one-form and the normal plane field one-form on a foliated Riemannian manifold

Part of: Global differential geometry

Published online by Cambridge University Press:  09 April 2009

Grant Cairns  and
Richard H. Escobales Jr
Grant Cairns
Affiliation:
School of Mathematics La Trobe UniversityMelbourne 3083Australia e-mail: [email protected]
Richard H. Escobales Jr
Affiliation:
Department of Mathematics Canisius CollegeBuffalo NY 14208USA e-mail: [email protected]
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Abstract

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For foliations on Riemannian manifolds, we develop elementary geometric and topological properties of the mean curvature one-form κ and the normal plane field one-form β. Through examples, we show that an important result of Kamber-Tondeur on κ is in general a best possible result. But we demonstrate that their bundle-like hypothesis can be relaxed somewhat in codimension 2. We study the structure of umbilic foliations in this more general context and in our final section establish some analogous results for flows.

Keywords

foliationmean curvature

MSC classification

Secondary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 62 , Issue 1 , February 1997 , pp. 46 - 63
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Abe, K., ‘Applications of a Riccati type differential equation to Riemannian manifolds with totally geodesic distributions’, Tôhoku Math. J. 25 (1973), 425–444.CrossRefGoogle Scholar
[2]Bochner, S., ‘Vector fields and Ricci curvature’, Bull. Amer. Math. Soc. 52 (1946), 776–797.CrossRefGoogle Scholar
[3]Borel, A., ‘Compact Clifford-Klein forms of symmetric spaces’, Topology 2 (1963), 111–122.CrossRefGoogle Scholar
[4]Bott, R. and Tu, L., Differential forms in algebraic topology (Springer, New York, 1982).CrossRefGoogle Scholar
[5]Brunella, M. and Ghys, E., ‘Umbilic foliations and transversely holomorphic flows’, J. Differential Geom. 41 (1995), 1–19.CrossRefGoogle Scholar
[6]Cairns, G., ‘A general description of totally geodesic foliations’, Tôhoku Math. J. 381 (1986), 37–55.Google Scholar
[7]Cairns, G., ‘A general description of totally geodesic foliations’, Michigan Math. J. 37 (1990), 145–159.Google Scholar
[8]Cairns, G. and Ghys, E., ‘Totally geodesic foliations on 4-manifolds’, J. Differential Geom. 23 (1986), 241–254.CrossRefGoogle Scholar
[9]Carrière, Y., ‘Appendix A- Variations on Riemannian flows’, in: Riemannian foliations (P. Molino) (Birkhauser, Boston, 1988) pp. 217–234.Google Scholar
[10]Domínguez, D., ‘Finiteness and tenseness theorems for Riemannian foliations’, preprint.Google Scholar
[11]Escobales, R., ‘The mean curvature cohomology class for foliations and the infinitesimal geometry of the leaves’, Differential Geom. Appl. 2 (1992), 167–178.CrossRefGoogle Scholar
[12]Escobales, R. and Parker, P., ‘Geometric consequences of the normal curvature cohomology class in umbilic foliations’, Indiana Univ. Math. J. 37 (1988), 389–408.CrossRefGoogle Scholar
[13]Gromoll, D. and Grove, K., ‘One-dimensional metric foliations in constant curvature spaces’, in: Differential geometry and complex analysis H. E. Rauch memorial volume (eds. Chavel, I. and Farkas, H. M.) (1985) pp. 165–168.CrossRefGoogle Scholar
[14]Haefliger, A., ‘Some remarks on foliations with minimal leaves’, J. Differential Geom. 15 (1980), 269–284.CrossRefGoogle Scholar
[15]Haefliger, A., ‘Groupoides d'holonomie et classifiants’, Astérisque 116 (1984), 98–107.Google Scholar
[16]Kamada, H., ‘Foliations on manifolds with positive constant curvature’, Tokyo J. Math. 16 (1993), 49–60.CrossRefGoogle Scholar
[17]Kamber, F. and Tondeur, Ph., ‘Foliations and metric’, in: Differential geometry (eds. Brooks, R., Gray, A., Reinhart, B.) (Birkhauser, Boston, 1983) pp. 103–152.Google Scholar
[18]Kamber, F. and Tondeur, Ph., ‘De Rham-Hodge theory for Riemannian foliations’, Math. Ann. 277 (1987), 415–431.CrossRefGoogle Scholar
[19]Masa, X., ‘Duality and minimality in Riemannian foliations’, Comment. Math. Helv. 67 (1992), 17–27.CrossRefGoogle Scholar
[20]Molino, P., Riemannian foliations (Birkhauser, Boston, 1988).CrossRefGoogle Scholar
[21]Nagano, T., ‘On fibred Riemann manifolds’, Sci. Papers College Gen. Ed. Univ. Tokyo 10 (1960), 17–27.Google Scholar
[22]O'Neill, B., ‘The fundamental equations of a submersion’, Michigan Math. J. 13 (1966), 459–469.CrossRefGoogle Scholar
[23]Oshikiri, G., ‘Mean curvature function of codimension-one foliations’, Comment. Math. Helv. 65 (1990), 79–84, II 66 (1991), 512–520.CrossRefGoogle Scholar
[24]Ranjan, A., ‘Structural equations and an integral formula for foliated manifolds’, Geom. Dedicata 20 (1986), 85–91.CrossRefGoogle Scholar
[25]Sergiescu, V., ‘Appendix B — Basic cohomology and tautness of Riemannian foliations’, in: Riemannian foliations (by Molino, P.) (Birkhauser, Boston, 1988) pp. 235–248.Google Scholar
[26]Sullivan, D., ‘Cycles for the dynamical study of foliated manifolds and complex manifolds’, Inventiones Math. 36 (1976), 225–255.CrossRefGoogle Scholar
[27]Sullivan, D., ‘A homological characterization of foliations consising of minimal surfaces’, Comment. Math. Helv. 54 (1979), 218–223.CrossRefGoogle Scholar
[28]Tondeur, P., Foliations on Riemannian manifolds (Springer, New York, 1988).CrossRefGoogle Scholar
[29]Tondeur, P., ‘Geometry of Riemannian foliations’, Seminar of mathematical sciences, 20 (Keio University, Yokohama, Japan, 1994).Google Scholar
[30]Walczak, P., ‘Foliations invariant under the mean curvature flow’, Illinois J. Math. 37 (1993), 609–623.CrossRefGoogle Scholar