Integral formulas for a foliated sub-Riemannian manifold

In this article, we deduce a series of integral formulas for a foliated sub-Riemannian manifold, which is a new geometric concept denoting a Riemannian manifold equipped with a distribution ${\mathcal D}$ and a foliation ${\mathcal F}$, whose tangent bundle is a subbundle of ${\mathcal D}$. Our integral formulas generalize some results for foliated Riemannian manifolds and involve the shape operators of ${\mathcal F}$ with respect to normals in ${\mathcal D}$ and the curvature tensor of induced connection on ${\mathcal D}$. The formulas also include arbitrary functions $f_j\ (0\le j<\dim{\mathcal F})$ depending on scalar invariants of the shape operators, and for a special choice of $f_j$ reduce to integral formulas with the Newton transformations of the shape operators. We apply our formulas to foliated sub-Riemannian manifolds with restrictions on the curvature and extrinsic geometry of ${\mathcal F}$ and to codimension-one foliations.