Integral formulas for a foliated sub-Riemannian manifold

Abstract

We apply the notion of foliation to a nonholonomic manifold, which was introduced for the geometric interpretation of constrained systems in mechanics. We prove a series of integral formulas for a foliated sub-Riemannian manifold, that is, a Riemannian manifold equipped with a distribution ${{\mathscr {D}}}$ and a foliation ${{\mathscr {F}}}$ whose tangent bundle is a subbundle of ${{\mathscr {D}}}$. Our integral formulas generalize some results for a foliated Riemannian manifold and involve the shape operators of ${\mathscr {F}}$ with respect to normals in ${\mathscr {D}}$, the curvature tensor of induced connection on ${\mathscr {D}}$ and arbitrary functions depending on elementary symmetric functions of eigenvalues of the shape operators. For a special choice of these functions, integral formulas with the Newton transformations of the shape operators of ${\mathscr {F}}$ are obtained. Application to a foliated sub-Riemannian manifold with restrictions on the curvature and extrinsic geometry of ${\mathscr {F}}$ and also to codimension-one foliations are given.