Jena 2013 – scientific programme

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GR: Fachverband Gravitation und Relativitätstheorie

GR 3: Hauptvorträge: Mathematische Methoden

GR 3.2: Invited Talk

Tuesday, February 26, 2013, 12:00–12:45, HS 6

How to reconstruct a metric by its unparameterized geodesics — •Vladimir Matveev — Mathematical Institute , University of Jena

We discuss whether it is possible to reconstruct a metric by its unparameterized geodesics, and how to do it effectively. We explain why this problem may be 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. The algorithm works very effectively if the searched metric is Ricci-flat. We also prove that almost every metric does not allow nontrivial geodesic equivalence, and construct all pairs of 4-dimensional geodesically equivalent metrics of Lorenz signature. If the time allows, I will also explain how this theory helped to solve two mathematical problems explicitly formulated by Sophus Lie in 1882, and the semi-Riemannian two-dimensional version of the projective Lichnerowicz-Obata conjecture. The new results of the talk are based on the papers arXiv:1010.4699, arXiv:1002.3934, arXiv:0806.3169, arXiv:0802.2344, arXiv:0705.3592 joint with Bryant, Bolsinov, Kiosak, Manno, Pucacco, and on an unpublished work with Trautman.