Lie contact structures and chains

TL;DR

This paper explores Lie contact structures, their equivalent descriptions, and the geometry of chains, revealing that chains encode the Lie contact structure and are not geodesics of any affine connection.

Contribution

It establishes the equivalence of Lie contact structures with split-quaternionic structures and analyzes the geometry of chains, showing their role in characterizing the underlying structure.

Findings

Chains are never geodesics of an affine connection.

The path geometry of chains uniquely determines the Lie contact structure.

Lie contact structures correspond to split-quaternionic structures on the contact distribution.

Abstract

Lie contact structures generalize the classical Lie sphere geometry of oriented hyperspheres in the standard sphere. They can be equivalently described as parabolic geometries corresponding to the contact grading of orthogonal real Lie algebra. It follows the underlying geometric structure can be interpreted in several equivalent ways. In particular, we show this is given by a split-quaternionic structure on the contact distribution, which is compatible with the Levi bracket. In this vein, we study the geometry of chains, a distinguished family of curves appearing in any parabolic contact geometry. Also to the system of chains there is associated a canonical parabolic geometry of specific type. Up to some exceptions in low dimensions, it turns out this can be obtained by an extension of the parabolic geometry associated to the Lie contact structure if and only if the latter is locally flat. In that case we can show that chains are never geodesics of an affine connection, hence, in particular, the path geometry of chains is always non-trivial. Using appropriately this fact, we conclude that the path geometry of chains allows to recover the Lie contact structure, hence, in particular, transformations preserving chains must preserve the Lie contact structure.