Integral formulas for a foliated sub-Riemannian manifold

TL;DR

This paper develops new integral formulas for foliated sub-Riemannian manifolds, generalizing existing results and involving shape operators and curvature tensors, with applications to specific geometric structures.

Contribution

It introduces a novel set of integral formulas for foliated sub-Riemannian manifolds, extending previous results to more general geometric contexts.

Findings

Formulas involving shape operators and curvature tensors derived.

Generalization of integral formulas for foliated Riemannian manifolds.

Applications to curvature restrictions and codimension-one foliations.

Abstract

In this article, we deduce a series of integral formulas for a foliated sub-Riemannian manifold, which is a new geometric concept denoting a Riemannian manifold equipped with a distribution ${\mathcal D}$ and a foliation ${\mathcal F}$, whose tangent bundle is a subbundle of ${\mathcal D}$. Our integral formulas generalize some results for foliated Riemannian manifolds and involve the shape operators of ${\mathcal F}$ with respect to normals in ${\mathcal D}$ and the curvature tensor of induced connection on ${\mathcal D}$. The formulas also include arbitrary functions $f_j\ (0\le j<\dim{\mathcal F})$ depending on scalar invariants of the shape operators, and for a special choice of $f_j$ reduce to integral formulas with the Newton transformations of the shape operators. We apply our formulas to foliated sub-Riemannian manifolds with restrictions on the curvature and extrinsic geometry of ${\mathcal F}$ and to codimension-one foliations.