Integral Formulas for a Foliation with a Unit Normal Vector Field

In this article, we prove integral formulas for a Riemannian manifold equipped with foliation F and unit vector field N orthogonal to F, generalize known (due Brito-Langevin-Rosenberg Andrzejewski-Walczak) foliations of codimension one. Our involve Newton transformations the shape operator respect curvature tensor induced connection on distribution D=TF⊕span(N), decomposition D can be regarded as codimension-one sub-Riemannian manifold. We apply our foliated (sub-)Riemannian manifolds restrictions extrinsic geometry foliation.