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Abstract: We discuss whether it is possible to reconstruct an affine connection, a (pseudo)-Riemannian metric or a Finsler metric
by 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, 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 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.