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Complex Manifolds

Ed. by Fino, Anna Maria

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Contact manifolds, Lagrangian Grassmannians and PDEs

Olimjon Eshkobilov
  • Dipartimento di Scienze Matematiche “G. L. Lagrange”, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy
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/ Gianni Manno
  • Corresponding author
  • Dipartimento di Scienze Matematiche “G. L. Lagrange”, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy
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/ Giovanni Moreno
  • Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warszawa, Poland
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/ Katja Sagerschnig
  • INDAM–Dipartimento di Scienze Matematiche “G. L. Lagrange”, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy
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Published Online: 2018-02-02 | DOI: https://doi.org/10.1515/coma-2018-0003


In this paper we review a geometric approach to PDEs. We mainly focus on scalar PDEs in n independent variables and one dependent variable of order one and two, by insisting on the underlying (2n + 1)-dimensional contact manifold and the so-called Lagrangian Grassmannian bundle over the latter. This work is based on a Ph.D course given by two of the authors (G. M. and G. M.). As such, it was mainly designed as a quick introduction to the subject for graduate students. But also the more demanding reader will be gratified, thanks to the frequent references to current research topics and glimpses of higher-level mathematics, found mostly in the last sections.

Keywords: Contact and symplectic manifolds; Jet spaces; Lagrangian Grassmannians; First and second order PDEs; Symmetries of PDEs; Characteristics; Monge-Ampère equations; PDEs on complex manifolds

MSC 2010: 32C15; 35A30; 35K96; 53C30; 53C55; 53D05; 53D10; 58A20; 58A30; 58J70


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About the article

Received: 2017-08-29

Accepted: 2018-01-18

Published Online: 2018-02-02

Citation Information: Complex Manifolds, Volume 5, Issue 1, Pages 26–88, ISSN (Online) 2300-7443, DOI: https://doi.org/10.1515/coma-2018-0003.

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© 2018 Olimjon Eshkobilov, et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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