Spinors in Five-Dimensional Contact Geometry

TL;DR

This paper develops a spinor-based differential geometric framework for five-dimensional contact structures, specifically G2 and Legendrean geometries, and computes their invariants including harmonic curvature and torsion.

Contribution

It introduces a novel spinor approach to analyze five-dimensional contact geometries and calculates explicit invariants for G2 contact structures.

Findings

Defined invariant directional derivatives in contact directions.

Constructed basic invariants such as harmonic curvature.

Calculated the invariant torsion for a specific G2 contact structure.

Abstract

We use classical (Penrose) two-component spinors to set up the differential geometry of two parabolic contact structures in five dimensions, namely $G_2$ contact geometry and Legendrean contact geometry. The key players in these two geometries are invariantly defined directional derivatives defined only in the contact directions. We explain how to define them and their usage in constructing basic invariants such as the harmonic curvature, the obstruction to being locally flat from the parabolic viewpoint. As an application, we calculate the invariant torsion of the $G_2$ contact structure on the configuration space of a flying saucer (always a five-dimensional contact manifold).