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Complex Manifolds

Ed. by Fino, Anna Maria

CiteScore 2017: 0.39

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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


  • [1] D. V. Alekseevskij, A. M. Vinogradov, and V. V. Lychagin. Basic ideas and concepts of differential geometry. In Geometry, I, volume 28 of Encyclopaedia Math. Sci., pages 1-264. Springer, Berlin, 1991.Google Scholar

  • [2] D. V. Alekseevsky, J. Gutt, G. Manno, and G. Moreno. Lowest degree invariant second-order PDEs over rational homogeneous contact manifolds. Commun. Contemp. Math.. https://doi.org/10.1142/S0219199717500894.CrossrefGoogle Scholar

  • [3] Dmitri Alekseevsky, Ricardo Alonso-Blanco, Gianni Manno, and Fabrizio Pugliese. Monge-Ampère equations on (para-)Kähler manifolds: from characteristic subspaces to special Lagrangian submanifolds. Acta Appl. Math., 120:3-27, 2012. ISSN 0167-8019. URL http://dx.doi.org/10.1007/s10440-012-9707-1.CrossrefGoogle Scholar

  • [4] Dmitri Alekseevsky, Ricardo Alonso-Blanco, Gianni Manno, and Fabrizio Pugliese. Finding solutions of parabolic monge-ampere equations by using the geometry of sections of the contact distribution. Differential Geometry and its Applications, 33, Supplement(0):144 - 161, 2014. ISSN 0926-2245. http://dx.doi.org/10.1016/j.difgeo.2013.10.015. URL http://www.sciencedirect.com/science/article/pii/S0926224513001034. The Interaction of Geometry and Representation Theory. Exploring new frontiers.Google Scholar

  • [5] Dmitri V. Alekseevsky, Ricardo Alonso-Blanco, Gianni Manno, and Fabrizio Pugliese. Contact geometry of multidimensional Monge-Ampère equations: characteristics, intermediate integrals and solutions. Ann. Inst. Fourier (Grenoble), 62 (2):497-524, 2012. ISSN 0373-0956. 10.5802/aif.2686. URL http://dx.doi.org/10.5802/aif.2686.CrossrefGoogle Scholar

  • [6] R. Alonso-Blanco, G. Manno, and F. Pugliese. Normal forms for Lagrangian distributions on 5-dimensional contact manifolds. Differential Geom. Appl., 27(2):212-229, 2009. ISSN 0926-2245. 10.1016/j.difgeo.2008.06.019. URL http://dx.doi.org/10.1016/j.difgeo.2008.06.019.Google Scholar

  • [7] V. I. Arnold. Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics. Springer, 1989. ISBN 3540968903,9783540968900.Google Scholar

  • [8] A. V. Bocharov, V. N. Chetverikov, S. V. Duzhin, N. G. Khor0kova, I. S. Krasil0shchik, A. V. Samokhin, Yu. N. Torkhov, A. M. Verbovetsky, and A. M. Vinogradov. Symmetries and conservation laws for differential equations of mathematical physics, volume 182 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1999. ISBN 0-8218-0958-X.Google Scholar

  • [9] R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffths. Exterior differential systems, volume 18 of Mathematical Sciences Research Institute Publications. Springer-Verlag, New York, 1991. ISBN 0-387-97411-3.Google Scholar

  • [10] Robert L. Bryant and Phillip A. Griffths. Characteristic cohomology of differential systems. II. Conservation laws for a class of parabolic equations. Duke Math. J., 78(3):531-676, 1995. ISSN 0012-7094. 10.1215/S0012-7094-95-07824-7. URL http://dx.doi.org/10.1215/S0012-7094-95-07824-7.Google Scholar

  • [11] Ch. Ehresmann. Introduction à la théorie des structures inffnitésimales et des pseudogroupes de Lie. In Colloque de Topologie et Géométrie Différentielle, Strasbourg, 1952, no. 11. La Bibliothèque Nationale et Universitaire de Strasbourg, 1953.Google Scholar

  • [12] Ch. Ehresmann. Introduction à la théorie des structures inffnitésimales et des pseudogroupes de Lie. In Géométrie Différentielle. Colloques Internationaux du Centre National de la Recherche Scientiffque, Strasbourg, 1953. Centre National de la Recherche Scientiffque, Paris, 1953.Google Scholar

  • [13] F. Reese Harvey and H. Blaine Lawson, Jr. Split special Lagrangian geometry. In Metric and differential geometry, volume 297 of Progr. Math., pages 43-89. Birkhäuser/Springer, Basel, 2012. 10.1007/978-3-0348-0257-4ff3. URL http://dx.doi.org/10.1007/978-3-0348-0257-4ff3.Google Scholar

  • [14] Reese Harvey and H. Blaine Lawson. Calibrated geometries. Acta Math., 148:47-157, 1982. 10.1007/BF02392726. URL http://dx.doi.org/10.1007/BF02392726.Google Scholar

  • [15] G. R. Jensen. Dupin hypersurfaces in Lie sphere geometry. ArXiv e-prints, May 2014.Google Scholar

  • [16] D. Joyce. Lectures on special Lagrangian geometry. ArXiv Mathematics e-prints, November 2001.Google Scholar

  • [17] Joseph Krasil’shchik and Alexander Verbovetsky. Geometry of jet spaces and integrable systems. J. Geom. Phys., 61(9): 1633-1674, 2011. ISSN 0393-0440. 10.1016/j.geomphys.2010.10.012. URL http://dx.doi.org/10.1016/j.geomphys.2010.10.012.Google Scholar

  • [18] Alexei Kushner, Valentin Lychagin, and Vladimir Rubtsov. Contact geometry and non-linear differential equations, volume 101 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2007. ISBN 978-0-521- 82476-7; 0-521-82476-1.Google Scholar

  • [19] Gianni Manno and Giovanni Moreno. Meta-symplectic geometry of 3rd order Monge-Ampère equations and their characteristics. SIGMA Symmetry Integrability Geom. Methods Appl., 12:032, 35 pages, 2016. ISSN 1815-0659. 10.3842/SIGMA.2016.032. URL http://dx.doi.org/10.3842/SIGMA.2016.032.Google Scholar

  • [20] Peter W. Michor. Manifolds of differentiable mappings, volume 3 of Shiva Mathematics Series. Shiva Publishing Ltd., Nantwich, 1980. ISBN 0-906812-03-8.Google Scholar

  • [21] Giovanni Moreno. The geometry of the space of cauchy data of nonlinear pdes. Central European Journal of Mathematics, 11(11):1960-1981, 2013. 10.2478/s11533-013-0292-y. URL http://arxiv.org/abs/1207.6290.Google Scholar

  • [22] Shigeyuki Morita. Geometry of differential forms. Translations of mathematical monographs, Iwanami series in modern mathematics 201. American Mathematical Society, 2001. ISBN 0821810456,9780821810453.Google Scholar

  • [23] Peter J. Olver. Applications of Lie groups to differential equations, volume 107 of Graduate Texts in Mathematics. Springer- Verlag, New York, second edition, 1993. ISBN 0-387-94007-3; 0-387-95000-1. 10.1007/978-1-4612-4350-2. URL http: //dx.doi.org/10.1007/978-1-4612-4350-2.Google Scholar

  • [24] Peter J. Olver. Equivalence, invariants, and symmetry. London Mathematical Society Le. Cambridge University Press, 1995. ISBN 0521478111,9780521478113.Google Scholar

  • [25] I. G. Petrovsky. Lectures on Partial Differential Equations. Dover Books on Mathematics. Dover Publications, 1992. ISBN 978-0486669021.Google Scholar

  • [26] Wulf Rossmann. Lie Groups: An Introduction through Linear Groups (Oxford Graduate Texts in Mathematics). Oxford University Press, 2006. ISBN 0199202516. URL https://www.amazon.com/Lie-Groups-Introduction-Graduate-Mathematics/dp/0199202516?SubscriptionId=0JYN1NVW651KCA56C102&tag=techkie-20&linkCode=xm2&camp=2025&creative=165953&creativeASIN=0199202516.Google Scholar

  • [27] Francesco Russo. On the geometry of some special projective varieties, volume 18 of Lecture Notes of the Unione Matematica Italiana. Springer, Cham; Unione Matematica Italiana, Bologna, 2016. ISBN 978-3-319-26764-7; 978-3- 319-26765-4. 10.1007/978-3-319-26765-4. URL http://dx.doi.org/10.1007/978-3-319-26765-4.CrossrefGoogle Scholar

  • [28] Gerd Rudolph; Matthias Schmidt. Differential geometry and mathematical physics. / Part I, Manifolds, lie groups and hamiltonian systems. Theoretical and mathematical physics (Springer (Firm)). Springer, 2013. ISBN 9789400753457,9400753454.Google Scholar

  • [29] Karen E. Smith, Lauri Kahanpää, Pekka Kekäläinen, and William Traves. An Invitation to Algebraic Geometry. Springer New York, 2000. 10.1007/978-1-4757-4497-2. URL https://doi.org/10.1007%2F978-1-4757-4497-2.Google Scholar

  • [30] D. The. Exceptionally simple PDE. ArXiv e-prints, March 2016.Google Scholar

  • [31] Dennis The. Conformal geometry of surfaces in the Lagrangian Grassmannian and second-order PDE. Proc. Lond. Math. Soc. (3), 104(1):79-122, 2012. ISSN 0024-6115. 10.1112/plms/pdr023. URL http://dx.doi.org/10.1112/plms/pdr023.Google Scholar

  • [32] Cédric Villani. Topics in optimal transportation, volume 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2003. ISBN 0-8218-3312-X. 10.1007/b12016. URL http://dx.doi.org/10.1007/b12016.Google Scholar

  • [33] Luca Vitagliano. Characteristics, Bicharacteristics, and Geometric Singularities of Solutions of PDEs. Int.J.Geom.Meth.Mod.Phys., 11(09):1460039, 2014. 10.1142/S0219887814600391.Google Scholar

  • [34] Keizo Yamaguchi. Contact geometry of higher order. Japan. J. Math. (N.S.), 8(1):109-176, 1982. ISSN 0289-2316.Google Scholar

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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