Homogeneous Integrable Legendrian Contact Structures in Dimension Five

Boris Doubrov,Dennis The,Alexandr Medvedev

doi:10.1007/s12220-019-00219-x

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.

Highlights

A Legendrian contact structure (M; E, F ) is defined to be a splitting of a contact distribution C into the direct sum of two subdistributions E, F that are maximally isotropic with respect to the naturally defined conformal symplectic structure on C
We assume that all our manifolds and related objects are complex analytic, many results are valid in the smooth category
We shall exclusively deal with integrable Legendrian contact structures, which means that both isotropic subdistributions are completely integrable

Summary

Introduction

A Legendrian contact structure (M; E, F ) is defined to be a splitting of a contact distribution C (on an odd-dimensional manifold M) into the direct sum of two subdistributions E, F that are maximally isotropic with respect to the naturally defined conformal symplectic structure on C. We classify ILC structures up to this duality and indicate which structures are self-dual, i.e. locally contact equivalent to their dual In his famous 1910 paper [5], Elie Cartan studied the geometry of rank two distributions on 5-manifolds having generic growth vector (2, 3, 5). For such structures, Cartan solved the local equivalence problem and obtained a classification of all multiply transitive models.. We integrate each of these structure equations and come up with the corresponding ILC model defined in terms of the system of 2nd order PDEs. in the Appendix we give the detailed Lie algebra isomorphisms establishing the correspondence between the Cartan equations of the reduced bundle and the model systems of 2nd order PDEs, the equivalence relations on the parameters and the duality

Legendrian contact structures

ILC structures in dimension five

Cartan analysis

Integration of structure equations