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 a previous result of mine. 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 Jacobowitz  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.

This is joint work with Taiji Marugame, Vladimir Matveev and Richard Montgomery.