An integral formula for a pair of singular distributions

TL;DR

This paper develops a new integral formula for pairs of singular distributions on Riemannian manifolds, extending classical results and providing tools for analyzing their geometric properties.

Contribution

It introduces a novel divergence theorem and derives a Codazzi equation for singular distributions, generalizing existing integral formulas in differential geometry.

Findings

Derived a divergence theorem for singular distributions

Established a Codazzi equation for pairs of singular distributions

Provided a generalized integral formula involving mixed scalar curvature

Abstract

The paper is devoted to differential geometry of singular distributions (i.e., of varying dimension) on a Riemannian manifold. Such distributions are defined as images of the tangent bundle under smooth endomorphisms. We prove the novel divergence theorem with the divergence type operator and deduce the Codazzi equation for a pair of singular distributions. Tracing our Codazzi equation yields expression of the mixed scalar curvature through invariants of distributions, which provides some splitting results. Applying our divergence theorem, we get the integral formula, generalizing the known one, with the mixed scalar curvature of a pair of transverse singular distributions.