A NEW PROOF OF FROBENIUS THEOREM AND APPLICATIONS

A proof of Frobenius theorem on local integrability of a given distrib ution on a finite or infinite dimensional manifold under weak differen tiability conditions is given using holonomy methods and the curvature two form of the associated connection. The local curvature two form, which measures the non-integrability of a given distribution, is studi ed and a variety of applications are given. The Inverse Problem in the Calculus of Variations appears as a particular case.