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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 Jean-Pierre Bourguignon for his 75th birthday
On the Convex Pfaff-Darboux Theorem of Ekeland and Nirenberg
Robert L. Bryant
Department of Mathematics, Duke University, PO Box 90320, Durham, NC 27708-0320, USA
Received July 20, 2023, in final form August 20, 2023; Published online August 23, 2023
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 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.)
Key words: Pfaff-Darboux theorem; convexity; utility theory.
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References
- Browning M., Chiappori P.-A., Efficient intra-household allocations: a general characterization and empirical tests, Econometrica 66 (1998), 1241-1278.
- Bryant R.L., Chern S.S., Gardner R.B., Goldschmidt H.L., Griffiths P.A., Exterior differential systems, Math. Sci. Res. Inst. Publ., Vol. 18, Springer, New York, 1991.
- Chiappori P.-A., Ekeland I., A convex Darboux theorem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 25 (1997), 287-297.
- Chiappori P.-A., Ekeland I., The economics and mathematics of aggregation: formal models of efficient group behavior, Found. Trends Microecon. 5 (2009), 1-151.
- Ekeland I., Nirenberg L., A convex Darboux theorem, Methods Appl. Anal. 9 (2002), 329-344.
- Zakalyukin V.M., Concave Darboux theorem, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), 633-638.
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