Seminarium KMMF "Teoria Dwoistości"

A Riemannian metric gives rise to geodesics. As unparameterised curves, at each point there is one geodesic in every direction. Does this arrangement of curves determine the metric and will any such arrangement of curves determine a metric? These are classical questions considered, for example, by Roger Liouville in 1889 who made considerable progress in answering these questions in two dimensions. This talk will be in two parts. Part 1 will discuss joint work with Robert Bryant and Maciej Dunajski: it solves the two-dimensional case. Part 2 will discuss joint work with Maciej Dunajski: it finds obstructions in the three-dimensional case.