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

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 $\mathcal{D}=T\mathcal{F}\oplus \mathrm{span}(N)$, where $\mathcal{F}$ is a foliation of $M$ and $N$ a unit vector field $g$-orthogonal to $\mathcal{F}$. Our integral formulas involve $r$th 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.

10 pages