Integral formulas for a foliated sub-Riemannian manifold

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