On the Convex Pfaff-Darboux Theorem of Ekeland and Nirenberg

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2023
• ISSN: 1815-0659
• Main Author: Bryant, Robert L.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2023
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/211969
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: On the Convex Pfaff-Darboux Theorem of Ekeland and Nirenberg. Robert L. Bryant. SIGMA 19 (2023), 060, 10 pages

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 ℝⁿ, 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 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 Sér. I Math. 327 (1998), 633-638], 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.)