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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: g.cairns@latrobe.edu.au
Richard H. Escobales Jr
Affiliation:
Department of Mathematics Canisius CollegeBuffalo NY 14208USA e-mail: escobalr@gort.canisius.edu
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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

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