Geodesic scattering on hyperboloids and Knoerrer’s map

Geodesic flow on ellipsoids is one of the most celebrated classical integrable systems considered by Jacobi in 1837. Moser revisited this problem in 1978 revealing the link with the modern theory of solitons. Surprisingly a similar question for hyperboloids did not get much attention, although the dynamics in this case is very different.
I will explain how to use the remarkable results of Moser and Knoerrer on the relations between Jacobi problem and integrable Neumann system on sphere to describe explicitly the geodesic scattering on hyperboloids. It will be shown also that Knoerrer's reparametrisation is closely related to the projectively equivalent metric on a quadric discovered in 1998 by Tabachnikov and, independently, by Matveev and Topalov, giving a new proof of their result. The projectively equivalent metric (in contrast to the usual one) turns out to be regular on the projective closure of hyperboloid, which allows us to extend Knoerrer's map to this closure.
The talk is based on a recent joint work with Lihua Wu.