On the Convex Pfaff-Darboux Theorem of Ekeland and Nirenberg

TL;DR

This paper generalizes the convex Pfaff-Darboux theorem, providing a broader local normal form for 1-forms with convexity conditions defined via torsion-free affine connections, extending previous results that required flat connections.

Contribution

It introduces a strengthened and generalized convex Pfaff-Darboux theorem applicable to Legendrian foliations with convexity defined by torsion-free affine connections.

Findings

Generalization of the convex Pfaff-Darboux theorem to non-flat affine connections

Extension of previous results to broader geometric settings

Provides necessary background and a new proof for the generalized theorem

Abstract

The classical Pfaff-Darboux theorem, which provides local 'normal forms' for $1$-forms on manifolds, has applications in the theory of certain economic models [Chiappori P.-A., Ekeland I., Found. Trends Microecon. 5 (2009), 1-151]. However, the normal forms needed in these models often come with an additional requirement of some type 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 $\mathbb{R}^n$, convexity has its usual meaning.) In [Methods Appl. Anal. 9 (2002), 329-344], Ekeland and Nirenberg were able to characterize necessary and sufficient conditions for a given 1-form $\omega$ to admit a convex local normal form (and to show that some earlier attempts [Chiappori P.-A., Ekeland I., Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 25 (1997), 287-297] and [Zakalyukin V.M., C. R. Acad. Sci. Paris S\'er. I Math. 327 (1998), 633-638] 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 of Ekeland and Nirenberg concerns the case in which the affine connection is flat.)