Homogeneous Integrable Legendrian Contact Structures in Dimension Five

  • Boris DoubrovEmail author
  • Alexandr Medvedev
  • Dennis The


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.


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

Mathematics Subject Classification

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



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).


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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

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