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 [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., \textit{C. R. Acad. Sci. Paris Sér. {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.)