A series of integral formulas for a foliated sub-Riemannian manifold

2021/03/03 by Vladimir Rovenski, Rovenski, Vladimir
Engineering · Mathematics · #53C12 #53C17 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Numerical methods in inverse problems #Thermoelastic and Magnetoelastic Phenomena

paper · pdf · doi:10.48550/arxiv.2103.02473

openalex publication_date 2021/03/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article, we prove a series of integral formulae for a codimension-one foliated sub-Riemannian manifold, i.e., a Riemannian manifold (M,g) equipped with a distribution \mathcal D=T\mathcal F⊕ \rm span(N), where \mathcal F is a foliation of M and N a unit vector field g-orthogonal to \mathcal F. Our integral formulas involve rth mean curvatures of \mathcal F, Newton transformations of the shape operator of \mathcal F with respect to N and the curvature tensor of induced connection on \mathcal D and generalize some known integral formulas (due to Brito-Langevin-Rosenberg, Andrzejewski-Walczak and the author) for codimension-one foliations. We apply our formulas to sub-Riemannian manifolds with restrictions on the curvature and extrinsic geometry of a foliation.

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