Legendrean and G2 Contact Structures

Abstract

We investigate two parabolic contact geometries: Legendrean contact structures and G2 contact structures. The methods used are mostly independent of the general theory of parabolic geometry. Building on the work of Eastwood and Nurowski we present a new method of generating G2 contact structures from five-dimensional Legendrean contact structures. This construction requires some input data, a choice of sections, and we calculate the minimal partial torsion, the obstruction to flatness of the G2 contact geometry, in terms of this input data. We show that in fact every G2 contact structure arises (locally) via this construction. Separately, inspired by the prolongation of the conformal-to-Einstein equation, we construct the standard tractor bundle for five-dimensional integrable Legendrean contact structures via prolongation of an invariantly defined partial di↵erential equation. We compute the partial curvature of the invariant partial connection on this bundle and show, by constructing an explicit isomorphism, that the geometry is locally isomorphic to the homogeneous model if and only if the partial curvature vanishes. We outline a similar prolongation procedure for G2 contact structures. There is a detailed review of relevant facts about contact manifolds, including the construction of the Rumin complex. We work out the required theory of partial connections on contact manifolds. We explain how to write many of the natural di↵erential operators on a contact manifold, for example, the Rumin complex, in terms of a suitably adapted partial connection.