Aerodynamics of flying saucers

TL;DR

This paper explores the geometric structures of flying saucers' configuration spaces in three-dimensional manifolds, revealing connections to contact, CR, and parabolic geometries, including cases with exceptional symmetries.

Contribution

It characterizes various geometric structures on the configuration space of flying saucers, linking them to advanced concepts like twistor, CR, and Cartan geometries, and identifies conditions for flatness and symmetry.

Findings

Configuration space C is always a five-dimensional contact manifold.

Different structures on M induce specific geometries on C, such as twistor and CR structures.

Examples of flat geometries with G2 symmetry are provided.

Abstract

We identify various structures on the configuration space C of a flying saucer, moving in a three-dimensional smooth manifold M. Always C is a five-dimensional contact manifold. If M has a projective structure, then C is its twistor space and is equipped with an almost contact Legendrean structure. Instead, if M has a conformal structure, then the saucer moves according to a CR structure on C. With yet another structure on M, the contact distribution in C is equipped with a cone over a twisted cubic. This defines a certain type of Cartan geometry on C (more specifically, a type of `parabolic geometry') and we provide examples when this geometry is `flat,' meaning that its symmetries comprise the split form of the exceptional Lie algebra G2.