Chains in CR geometry as geodesics of a Kropina metric

TL;DR

This paper demonstrates that chains in CR geometry can be viewed as geodesics of a specific Kropina metric, enabling new geometric and variational methods to analyze CR structures and chains.

Contribution

It establishes a novel link between CR chains and Kropina geodesics, studies projective equivalence of Kropina metrics, and extends chain connectivity results to global settings.

Findings

Chains are geodesics of a constructed Kropina metric.

Two projectively equivalent Kropina metrics with non-integrable kernels are trivially equivalent.

Any two points in certain CR manifolds can be connected by a chain.

Abstract

With the help of a generalization of the Fermat principle in general relativity, we show that chains in CR geometry are geodesics of a certain Kropina metric constructed from the CR structure. We study the projective equivalence of Kropina metrics and show that if the kernel distributions of the corresponding 1-forms are non-integrable then two projectively equivalent metrics are trivially projectively equivalent. As an application, we show that sufficiently many chains determine the CR structure up to conjugacy, generalizing and reproving the main result of [J.-H. Cheng, 1988]. The correspondence between geodesics of the Kropina metric and chains allows us to use the methods of metric geometry and the calculus of variations to study chains. We use these methods to re-prove the result of [H. Jacobowitz, 1985] that locally any two points of a strictly pseudoconvex CR manifolds can be joined by a chain. Finally, we generalize this result to the global setting by showing that any two points of a connected compact strictly pseudoconvex CR manifold which admits a pseudo-Einstein contact form with positive Tanaka-Webster scalar curvature can be joined by a chain.