Integral Formulas for a Foliation with a Unit Normal Vector Field

Vladimir Rovenski
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Vladimir Rovenski: Department of Mathematics, University of Haifa, Mount Carmel, Haifa 3498838, Israel

Mathematics, 2021, vol. 9, issue 15, 1-11

Abstract: In this article, we prove integral formulas for a Riemannian manifold equipped with a foliation F and a unit vector field N orthogonal to F , and generalize known integral formulas (due to Brito-Langevin-Rosenberg and Andrzejewski-Walczak) for foliations of codimension one. Our integral formulas involve Newton transformations of the shape operator of F with respect to N and the curvature tensor of the induced connection on the distribution D = T F ? span ( N ) , and this decomposition of D can be regarded as a codimension-one foliation of a sub-Riemannian manifold. We apply our formulas to foliated (sub-)Riemannian manifolds with restrictions on the curvature and extrinsic geometry of the foliation.

Keywords: Riemannian manifold; foliation; harmonic distribution; Newton transformation; shape operator; r -th mean curvature (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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