Homogeneous integrable Legendrian contact structures in dimension five

TL;DR

This paper classifies five-dimensional integrable Legendrian contact structures using parabolic geometry, providing explicit curvature formulas and a Petrov classification based on harmonic curvature root types.

Contribution

It offers a complete local classification of transitive five-dimensional integrable Legendrian contact structures with symmetry, using explicit curvature analysis and Petrov classification.

Findings

Harmonic curvature vanishes iff the PDE system is trivializable.

Binary quartic form classifies structures via Petrov types.

Complete local classification for structures with transitive symmetry.

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.