Twistor Theory of Dancing Paths

Abstract

Given a path geometry on a surface 𝒰, we construct a causal structure on a four-manifold which is the configuration space of non-incident pairs (point, path) on 𝒰. This causal structure corresponds to a conformal structure if and only if 𝒰 is a real projective plane, and the paths are lines. We give the example of the causal structure given by a symmetric sextic, which corresponds to an SL(2, ℝ)-invariant projective structure where the paths are ellipses of area π centred at the origin. We shall also discuss a causal structure on a seven-dimensional manifold corresponding to non-incident pairs (point, conic) on a projective plane.