Frobenius theorem on Pfaffian systems

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A theorem on the conditions for a system of Pfaffian equations to be completely integrable, or (in geometrical terms) on conditions under which a given field of $n$-dimensional tangent subspaces on a differentiable manifold is the tangent field of some foliation. For several equivalent formulations of the Frobenius theorem, see the articles Involutive distribution; Cauchy problem; for a version with minimum differentiability requirements see [2]. The name of the theorem is connected with the account of it in [1], but does not accord with the information given there about its history.


[1] G. Frobenius, "Ueber das Pfaffsche Problem" J. Reine Angew. Math. , 82 (1877) pp. 230–315
[2] P. Hartman, "Ordinary differential equations" , Birkhäuser (1982) MR0658490 Zbl 0476.34002


G. Frobenius actually also treats the normal form of a differential form of degree 1.

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Frobenius theorem on Pfaffian systems. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article