A Frobenius Theorem for Corank-1 Continuous Distributions in Dimensions two and three

TL;DR

This paper extends Frobenius' classical theorem to continuous distributions of corank-1 in dimensions two and three, establishing conditions under which such distributions are integrable.

Contribution

It introduces a notion of asymptotic involutivity for continuous distributions and proves it implies integrability in low dimensions, generalizing Frobenius' theorem.

Findings

Asymptotic involutivity implies integrability in dimensions ≤3

Generalization of Frobenius theorem to continuous distributions

Unique integrability under the new conditions

Abstract

We formulate a notion of (uniform) asymptotic involutivity and show that it implies (unique) integrability of corank-1 continuous distributions in dimensions three or less. This generalizes and extends a classical theorem of Frobenius Theorem which says that an involutive C^1 distribution is uniquely integrable.