Canonical curves and Kropina metrics in Lagrangian contact geometry

TL;DR

This paper explores the relationship between Lagrangian contact structures, conformal structures, and Kropina metrics, revealing how chains and null-chains relate to geodesics and how these structures can be uniquely determined.

Contribution

It introduces a Fefferman-type construction linking Lagrangian contact and conformal structures, and demonstrates how chains are realized as geodesics of Kropina metrics, with implications for structure rigidity.

Findings

Chains are projections of null-geodesics in Fefferman space.

Chains can be realized as geodesics of Kropina metrics.

Sufficiently many chains determine the Lagrangian contact structure.

Abstract

We present a Fefferman-type construction from Lagrangian contact to conformal structures and examine several related topics. In particular, we concentrate on describing the canonical curves and their correspondence. We show that chains and null-chains of an integrable Lagrangian contact structure are the projections of null-geodesics of the Fefferman space. Employing the Fermat principle, we realize chains as geodesics of Kropina (pseudo-Finsler) metrics. Using recent rigidity results, we show that ``sufficiently many'' chains determine the Lagrangian contact structure. Separately, we comment on Lagrangian contact structures induced by projective structures and the special case of dimension three.