Cone structures and path geometries with constant torsion

TL;DR

This paper explores three-dimensional path geometries with constant torsion, linking them to cone structures on ruled surfaces and integrable systems with special Lax pairs, advancing understanding of their geometric and algebraic properties.

Contribution

It introduces the concept of constant torsion in path geometries and establishes a correspondence with cone structures modeled on homogeneous ruled surfaces, along with integrable system descriptions.

Findings

Path geometries with constant torsion correspond to specific cone structures.

New integrable systems with dispersionless, non-isospectral Lax pairs are identified.

A classification of these geometries and their algebraic models is provided.

Abstract

We study three-dimensional path geometries with nontrivial torsion of maximal rank. We introduce the notion of constant torsion and show that such path geometries are in one-to-one correspondence with certain cone structures modeled on homogeneous ruled surfaces. We also describe these cone structures in terms of solutions to new integrable systems admitting dispersionless, non-isospectral Lax pairs.