A general Frobenius' Theorem via the Transport of Currents

TL;DR

This paper generalizes Frobenius' Theorem to include singular vector measures, connecting classical differential geometry with PDEs and Magnetohydrodynamics, and extends the understanding of vector field commutation in complex settings.

Contribution

It extends Frobenius' Theorem to vector fields involving singular measures using PDE analysis and links classical geometry with MHD results.

Findings

Generalization of Frobenius' Theorem to measure-valued vector fields

Connection between geometric transport equations and classical theorems

Interpretation of Alfvén's theorem as a time-dependent Frobenius' Theorem

Abstract

A classical result in Differential Geometry states that the flows of two smooth vector fields commute if and only if their Lie Bracket vanishes. In this work, we extend this result to a more general setting where one of the vector fields is bounded and Lipschitz, while the other may be a singular vector-valued measure, i.e. a normal 1-current. This result is achieved via the study of two distinct evolutionary PDEs describing the transport of vector quantities (the Vector Advection Equation and the Geometric Transport Equation). Furthermore, we show that a celebrated theorem by Alfv\'en in Magnetohydrodynamics can be interpreted as a suitable time-dependent version of Frobenius' Theorem. Our approach builds on recent advances concerning the Geometric Transport Equation for currents [5, 6].