Advertisement

Homogeneous Integrable Legendrian Contact Structures in Dimension Five

  • Boris DoubrovEmail author
  • Alexandr Medvedev
  • Dennis The
Article
  • 13 Downloads

Abstract

We consider Legendrian contact structures on odd-dimensional complex analytic manifolds. We are particularly interested in integrable structures, which can be encoded by compatible complete systems of second order PDEs on a scalar function of many independent variables and considered up to point transformations. Using the techniques of parabolic differential geometry, we compute the associated regular, normal Cartan connection and give explicit formulas for the harmonic part of the curvature. The PDE system is trivializable by means of point transformations if and only if the harmonic curvature vanishes identically. In dimension five, the harmonic curvature takes the form of a binary quartic field, so there is a Petrov classification based on its root type. We give a complete local classification of all five-dimensional integrable Legendrian contact structures whose symmetry algebra is transitive on the manifold and has at least one-dimensional isotropy algebra at any point.

Keywords

Legendrian structures Symmetry algebra Curvature module Multiply transitive Complete systems of PDEs 

Mathematics Subject Classification

Primary: 58J70 Secondary: 35A30 53A40 53B15 53D10 22E46 

Notes

Acknowledgements

The Cartan and DifferentialGeometry packages in Maple (written by Jeanne Clelland and Ian Anderson respectively) provided an invaluable framework for implementing the Cartan reduction method and subsequently carrying out the analysis of the structures obtained. The work of the second and third authors was supported by ARC Discovery Grants DP130103485 and DP110100416 respectively. D.T. was also supported by Project M1884-N35 of the Austrian Science Fund (FWF).

References

  1. 1.
    Baston, R.J., Eastwood, M.G.: The Penrose Transform: Its Interaction with Representation Theory, Oxford Mathematical Monographs. Clarendon Press, Oxford (1989)zbMATHGoogle Scholar
  2. 2.
    Bol, G.: Über topologische Invarianten von zwei Kurvenscharen in Raum. Abhandlungen Math. Sem. Univ. Hamburg. 9(1), 15–47 (1932)CrossRefzbMATHGoogle Scholar
  3. 3.
    Čap, A.: Correspondence spaces and twistor spaces for parabolic geometries. J. Reine Angew. Math. 582, 143–172 (2005)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Čap, A., Slovák, J.: Parabolic Geometries I: Background and General Theory, Mathematical Surveys and Monographs, vol. 154. American Mathematical Society, Providence (2009)zbMATHGoogle Scholar
  5. 5.
    Cartan, É.: Les systèmes de Pfaff à cinq variables et les équations aux dérivées partielles du second ordre. Ann. Sci. Éc. Norm. Supér. (3) 27, 109–192 (1910)CrossRefzbMATHGoogle Scholar
  6. 6.
    Cartan, É.: Sur les variétes à connexion projective. Bull. Soc. Math. Fr. 52, 205–241 (1924)CrossRefzbMATHGoogle Scholar
  7. 7.
    Cartan, É.: Sur la géométrie pseudo-conforme des hypersurfaces de l’espace de deux variables complexes. Ann. Mat. Pura Appl. 11, 17–90 (1932)CrossRefzbMATHGoogle Scholar
  8. 8.
    Cartan, É.: Sur la géométrie pseudo-conforme des hypersurfaces de l’espace de deux variables complexes II. Ann. Sci. Norm. Super. Pisa 4, 333–354 (1932)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Cartan, É.: Sur les domaines bornés homogènes de l’espace de \(n\) variables complexes. Abh. Math. Sere. Hamburg 11/12, 116–162 (1935)CrossRefzbMATHGoogle Scholar
  10. 10.
    Doubrov, B., Govorov, A.: A new example of a generic 2-distribution on a 5-manifold with large symmetry algebra, arXiv:1305.7297 (2013)
  11. 11.
    Doubrov, B., Medvedev, A., The, D.: Homogeneous Levi non-degenerate hypersurfaces in \(\mathbb{C}^3\), arXiv:1711.02389 (2017)
  12. 12.
    Gardner, R.B.: The method of equivalence and its applications, CBMS-NSF Regional Conference Series in Applied Mathematics, 58. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1989)Google Scholar
  13. 13.
    Gaussier, H., Merker, J.: Symmetries of partial differential equations. J. Korean Math. Soc. 40(3), 517–561 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kaneyuki, S., Tsuji, T.: Classification of homogeneous bounded domains of lower dimension. Nagoya Math. J. 53, 1–46 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kostant, B.: Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. Math. 74(2), 329–387 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kruglikov, B., The, D.: The gap phenomenon in parabolic geometries. J. Reine Angew. Math. 723, 153–215 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Loboda, A.V.: Homogeneous real hypersurfaces in \({\mathbb{C}}^3\) with two-dimensional isotropy groups, Tr. Mat. Inst. Steklova 235 (2001), Anal. i Geom. Vopr. Kompleks. Analiza, 114–142; (Russian) translation in Proc. Steklov Inst. Math. 2001, no. 4 (235), 107–135Google Scholar
  18. 18.
    Loboda, A.V.: Homogeneous strictly pseudoconvex hypersurfaces in \({\mathbb{C}}^3\) with two-dimensional isotropy groups. Mat. Sb. 192(12), 3–24 (2001); (Russian) translation in Sb. Math. 192 (2001), no. 11–12, 1741–1761Google Scholar
  19. 19.
    Loboda, A.V.: Determination of a homogeneous strictly pseudoconvex surface from the coefficients of its normal equation. Matematicheskie Zametki 73(3), 453–456 (2003); (Russian) translation in Mathematical Notes, vol. 73, no. 3, 2003, pp. 419–423Google Scholar
  20. 20.
    Merker, J.: Lie symmetries and CR geometry. Complex analysis. J. Math. Sci. (N. Y.) 154(6), 817–922 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Olver, P.J.: Symmetry, invariants, and equivalence. Springer, New York (1995)CrossRefzbMATHGoogle Scholar
  22. 22.
    Pyatetskii-Shapiro, I.I.: On a problem proposed by E. Cartan. Dokl. Akad. Nauk SSSR 124, 272–273 (1959)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Sukhov, A.: Segre varieties and Lie symmetries. Math. Z. 238(3), 483–492 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sukhov, A.: On transformations of analytic CR-structures. Izv. Math. 67(2), 303–332 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Takeuchi, M.: Lagrangean contact structures on projective cotangent bundles. Osaka J. Math. 31(4), 837–860 (1994)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Tresse, A.: Détermination des invariants ponctuels de l’équation différentielle ordinaire du second ordre \(y^{\prime \prime } = \omega (x,y,y^{\prime })\), Leipzig. 87 S. gr. \(8^\circ \) (1896)Google Scholar
  27. 27.
    Winkelmann, J.: The Classification of 3-Dimensional Homogeneous Complex Manifolds. Lecture Notes in Math, vol. 1602. Springer, New York (1995)CrossRefzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Faculty of Mathematics and MechanicsBelarusian State UniversityMinskBelarus
  2. 2.School of Science and TechnologyUniversity of New EnglandArmidaleAustralia
  3. 3.International School for Advanced StudiesTriesteItaly
  4. 4.Mathematical Sciences InstituteAustralian National UniversityCanberraAustralia
  5. 5.Fakultät für MathematikUniversität WienWienAustria
  6. 6.Department of Mathematics and Statistics, Faculty of Science and TechnologyUiT The Arctic University of NorwayTromsøNorway

Personalised recommendations