### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 031, 19 pages      arXiv:2201.13048      https://doi.org/10.3842/SIGMA.2022.031
Contribution to the Special Issue on Twistors from Geometry to Physics in honour of Roger Penrose

### Spinors in Five-Dimensional Contact Geometry

Michael Eastwood a and Timothy Moy b
a) School of Mathematical Sciences, University of Adelaide, SA 5005, Australia
b) Clare College, University of Cambridge, CB2 1TL, England, UK

Received January 31, 2022, in final form April 13, 2022; Published online April 16, 2022

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).

Key words: spinors; contact geometry; parabolic geometry.

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