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 com...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2023 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2023
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/211969 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On the Convex Pfaff-Darboux Theorem of Ekeland and Nirenberg. Robert L. Bryant. SIGMA 19 (2023), 060, 10 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862719358214602752 |
|---|---|
| author | Bryant, Robert L. |
| author_facet | Bryant, Robert L. |
| citation_txt | On the Convex Pfaff-Darboux Theorem of Ekeland and Nirenberg. Robert L. Bryant. SIGMA 19 (2023), 060, 10 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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.)
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| first_indexed | 2026-03-20T23:29:09Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211969 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-20T23:29:09Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bryant, Robert L. 2026-01-20T16:10:56Z 2023 On the Convex Pfaff-Darboux Theorem of Ekeland and Nirenberg. Robert L. Bryant. SIGMA 19 (2023), 060, 10 pages 1815-0659 2020 Mathematics Subject Classification: 58A15; 91B16 arXiv:1512.07100 https://nasplib.isofts.kiev.ua/handle/123456789/211969 https://doi.org/10.3842/SIGMA.2023.060 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.) This article is dedicated to Jean-Pierre Bourguignon, with much admiration, on the occasion of his 75th birthday. Thanks to Duke University for its support via a research grant and to the National Science Foundation for its support via DMS-9870164 (during which most of the research for this article was done) and DMS-1359583 (during which this article was written). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On the Convex Pfaff-Darboux Theorem of Ekeland and Nirenberg Article published earlier |
| spellingShingle | On the Convex Pfaff-Darboux Theorem of Ekeland and Nirenberg Bryant, Robert L. |
| title | On the Convex Pfaff-Darboux Theorem of Ekeland and Nirenberg |
| title_full | On the Convex Pfaff-Darboux Theorem of Ekeland and Nirenberg |
| title_fullStr | On the Convex Pfaff-Darboux Theorem of Ekeland and Nirenberg |
| title_full_unstemmed | On the Convex Pfaff-Darboux Theorem of Ekeland and Nirenberg |
| title_short | On the Convex Pfaff-Darboux Theorem of Ekeland and Nirenberg |
| title_sort | on the convex pfaff-darboux theorem of ekeland and nirenberg |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211969 |
| work_keys_str_mv | AT bryantrobertl ontheconvexpfaffdarbouxtheoremofekelandandnirenberg |