How to reconstruct a metric by its unparameterized geodesics

Abstract:

We  discuss whether it is possible to reconstruct an affine  connection, a (pseudo)-Riemannian  metric or a Finsler metricby its unparameterized geodesics, and how to do it effectively. We explain why this problem is interesting for general relativity.  We show how to understand whether all curves from a sufficiently big family are unparameterized geodesics of a certain affine connection, and how to reconstruct algorithmically a generic 4-dimensional metric by its unparameterized  geodesics.  I will also explain how this theory helped to solve two problems explicitly formulated by Sophus Lie in 1882. This  portion of results  is joint with R.  Bryant,  A. Bolsinov,  V.  Kiosak, G. Manno, G. Pucacco.

At the end of my talk,  if the time allows (which is usually not the case), I will explain that the so-called chains in the CR-geometry are geodesics of a so-called Kropina  Finsler metric.  I will show that sufficiently many geodesics determine the Kropina Finsler metric, which re-proves and generalizes the famous result of Jih-Hsin Cheng, 1988,  that chains dermine the CR structure.  This  correspondence  between chains and Kropina geodesics  allows us to use the methods of metric geometry  to study chains,  we employ it  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, and generalize it to a global setting.   This portion of results is joint with J.-H. Cheng, T. Marugame,  R. Montgomery.