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

Published online by Cambridge University Press:  09 April 2009

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.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1997

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