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

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