A Frobenius Theorem on Fr\'echet Manifolds

TL;DR

This paper proves a Frobenius theorem for Fréchet manifolds, establishing conditions under which tangent distributions are integrable and lead to foliations, using a variational approach and differential forms.

Contribution

It introduces Condition W for split tangent subbundles and proves that involutivity plus this condition guarantees integrability on Fréchet manifolds.

Findings

Involutivity and Condition W imply integrability of tangent distributions.

Existence of a unique maximal foliation for integrable distributions.

Dual formulation characterizes integrability via exterior derivatives of annihilators.

Abstract

We investigate the integrability of Fr\'{e}chet tangent distributions on Fr\'{e}chet manifolds. We introduce the local well-posedness Condition W for split tangent subbundles, which reduces the local integrability problem to solving initial value problems with parameters whose solutions define curves tangent to the distribution. By applying a variational approach to establish the existence and uniqueness of these solutions, we prove a Frobenius theorem stating that involutivity and Condition W are sufficient for integrability. This yields the existence of a unique maximal foliation of the manifold. Furthermore, we provide a dual formulation of the theorem using differential forms, which characterizes the algebraic conditions for integrability via the exterior derivative of the subbundle's local annihilator.