How to reconstruct a metric by its unparameterised geodesics

Host Institution:

La Trobe University

Title of Seminar:

How to reconstruct a metric by its unparameterised geodesics

Speaker's Name:

Prof Vladimir Matveev

Speaker's Institution:

University of Jena, Germany

Time and Date:

Monday 26 September, 2011, 2:00 PM AEST

Seminar Abstract:

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 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 unparameterised geodesics. The algorithm works most effectively if the 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 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 and arXiv:0705.3592;  joint with Bryant, Bolsinov,  Kiosak, Manno and Pucacco.

Seminar Convenor:

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