On the Convex Pfaff-darboux Theorem of Ekeland and Nirenberg

The classical Pfaff-Darboux Theorem, which provides local ‘normal forms’ for 1-forms on manifolds, has applications in the theory of certain economic models [3]. However, the normal forms needed in these models come with an additional requirement of convexity, which is not provided by the classical proofs of the Pfaff-Darboux Theorem. (The appropriate notion of ‘convexity’ is a feature of the economic model. In the simplest case, when the economic model is formulated in a domain in R, convexity has its usual meaning.) In [4], Ekeland and Nirenberg were able to characterize necessary and sufficient conditions for a given 1-form ω to admit a convex local normal form (and to show that some earlier attempts [2, 5] at this characterization had been unsuccessful). In this article, after providing some necessary background, I prove a strengthened and generalized convex Pfaff-Darboux Theorem, one that covers the case of a Legendrian foliation in which the notion of convexity is defined in terms of a torsion-free affine connection on the underlying manifold. (The main result in [4] concerns the case in which the affine connection is flat.)