Integrability of normal distributions Part 2: Neat foliations by manifolds with boundary

TL;DR

This paper completes the theoretical framework for neatly integrable normal distributions on manifolds with boundary, establishing a correspondence with neat foliations by manifolds with boundary, extending classical foliation theory.

Contribution

It extends the theory of integrable distributions to manifolds with boundary, introducing neat foliations with leaves that may vary in dimension and have boundaries contained in the ambient boundary.

Findings

Established a one-to-one correspondence between neatly integrable normal distributions and neat foliations.

Extended classical foliation theory to manifolds with boundary.

Characterized leaves as weakly embedded submanifolds with boundary.

Abstract

This paper completes the foundations of neatly integrable normal distribution theory on manifolds with boundary. Normal distributions are those which contain vectors transverse to the boundary along its entirety. The theory is observed to be entirely analogous with the theory of integrable distributions on manifolds due to Stefan and Sussmann. The main result is a one-to-one correspondence between so-called neatly integrable normal distributions and neat foliations by manifolds with boundary. Neat foliations are allowed to have non-constant dimension and the leaves have boundary contained in the ambient boundary. The leaves satisfy a characteristic property formally identical to that of weakly embedded submanifolds except in the category of manifolds with boundary.