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arXiv · 2604.22472

A Frobenius Theorem on Fr\'echet Manifolds

Kaveh Eftekharinasab
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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.

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Kaveh Eftekharinasab. 2026-04-24. A Frobenius Theorem on Fr\'echet Manifolds. https://arxiv.org/abs/2604.22472

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