# How to reconstruct a metric by its unparametrized geodesics

# How to reconstruct a metric by its unparametrized geodesics

We discuss whether it is possible to reconstruct an affine connection, a (pseudo)-Riemannian metric or a Finsler metric by its unparametrized 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 unparametrized geodesics of a certain affine connection, and how to reconstruct algorithmically a generic 4-dimensional metric by its unparametrized 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, 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 reproves and generalizes the famouse result of Jih-Hsin Cheng, 1988, that chains determine 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.