A simple proof of Frobenius’s integration theorem

Abstract

The connection between Stokes's Integral Theorem and the Frobenius-Cartan Integration Theorem concerning Pfaffian systems has been noted a long time. In this note, we generalize Stokes's theorem to implicit vector valued differential forms and derive from it a general Frobenius theorem concerning mappings in Banach spaces. The only difficulty in the proof arises in the need to show differentiability with respect to a parameter of solutions of a certain differential equation, but is is easily overcome. The generality of the theorem seems to be necessary for applications to the new subjects of infinite groups and of differential geometry in infinitely many dimensions. E.g., it allows us to associate a local group to any infinite-dimensional Lie algebra in a Banach space. For finite dimensional vector spaces we obtain the classical theorem with nearly minimal differentiability conditions [4]. Also for finite dimensional spaces, one might derive from it parts of the Cartan-Kahler theory of integral manifolds [3 ] for not completely integrable CI systems.