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Vol. 52 No. 3 (2026)
Serdica Mathematical Journal

Articles

A Frobenius Theorem on Fréchet Manifolds

https://doi.org/10.55630/serdica.2026.52.269-294

Published 25-09-2026

  • Kaveh Eftekharinasab

Kaveh Eftekharinasab
Institute of Mathematics of the National Academy of Sciences of Ukraine

Abstract

We investigate the integrability of Fréchet tangent distributions on Fréchet 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.

2020 Mathematics Subject Classification:

58A30, 58B10, 58E05, 57R30

Keywords

Frobenius theorem, foliations, Palais–Smale condition, Fréchet manifolds, Keller's \(C_{c}^{k}\)-calculus, involutive distributions

Full text

Author Details

Kaveh Eftekharinasab

Algebra and Topology Department
Institute of Mathematics of the National Academy of Sciences of Ukraine
Tereshchenkivska st. – 3, 01024, Kyiv, Ukraine
e-mail: kaveh@imath.kiev.ua

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