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Integral Formulas for a Foliation with a Unit Normal Vector Field

Author

Listed:
  • Vladimir Rovenski

    (Department of Mathematics, University of Haifa, Mount Carmel, Haifa 3498838, Israel)

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.

Suggested Citation

  • Vladimir Rovenski, 2021. "Integral Formulas for a Foliation with a Unit Normal Vector Field," Mathematics, MDPI, vol. 9(15), pages 1-11, July.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:15:p:1764-:d:601622
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