
SIGMA 19 (2023), 060, 10 pages arXiv:1512.07100
https://doi.org/10.3842/SIGMA.2023.060
Contribution to the Special Issue on Differential Geometry Inspired by Mathematical Physics in honor of JeanPierre Bourguignon for his 75th birthday
On the Convex PfaffDarboux Theorem of Ekeland and Nirenberg
Robert L. Bryant
Department of Mathematics, Duke University, PO Box 90320, Durham, NC 277080320, USA
Received July 20, 2023, in final form August 20, 2023; Published online August 23, 2023
Abstract
The classical PfaffDarboux 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), 1151]. 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 PfaffDarboux 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), 329344], 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), 287297] and [Zakalyukin V.M., C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), 633638] at this characterization had been unsuccessful). In this article, after providing some necessary background, I prove a strengthened and generalized convex PfaffDarboux theorem, one that covers the case of a Legendrian foliation in which the notion of convexity is defined in terms of a torsionfree affine connection on the underlying manifold. (The main result of Ekeland and Nirenberg concerns the case in which the affine connection is flat.)
Key words: PfaffDarboux theorem; convexity; utility theory.
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References
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 Chiappori P.A., Ekeland I., A convex Darboux theorem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 25 (1997), 287297.
 Chiappori P.A., Ekeland I., The economics and mathematics of aggregation: formal models of efficient group behavior, Found. Trends Microecon. 5 (2009), 1151.
 Ekeland I., Nirenberg L., A convex Darboux theorem, Methods Appl. Anal. 9 (2002), 329344.
 Zakalyukin V.M., Concave Darboux theorem, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), 633638.

