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.