Integral formulas for a foliated sub-Riemannian manifold (Q2688845)

The authors apply the notion of foliation to a nonholonomic manifold, whichwas introduced for the geometric interpretation of constrained systems in mechanics. The authors prove a series of integral formulas for a foliated sub-Riemannian manifold, that is, a Riemannian manifold equipped with a distribution \(D\) and a foliation \(F\) whose tangent bundle is a subbundle of \(D\). Our integral formulas generalize some results for a foliated Riemannian manifold and involve the shape operators of \(F\) with respect to normals in \(D\), the curvature tensor of induced connection on \(D\) 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 \(F\) are obtained. Application to a foliated sub-Riemannian manifold with restrictions on the curvature and extrinsic geometry of \(F\) and also to codimensionone foliations are given.