Geodesic scattering on hyperboloids and Knörrer’s map

Abstract

We use the results of Moser and Knörrer on relations between geodesics on quadrics and solutions of the classical Neumann system to describe explicitly the geodesic scattering on hyperboloids. We explain the relation of Knörrer’s reparametrisation with projectively equivalent metrics on quadrics introduced by Tabachnikov and independently by Matveev and Topalov, giving a new proof of their result. We show that the projectively equivalent metric is regular on the projective closure of hyperboloids and extend Knörrer’s map to this closure.

Integrable systems on the sphere, ellipsoid and hyperboloid

“…There is also an intermediate case when geodesics stuck spiralling around the neck, see the possible scattering on a picture from [43], see Fig. 3: In this note, we have taken a simple preparatory step to study the problem of geodesic scattering on the hyperboloids associated with such classical problems as Kowalevsky top, Goryachev-Chaplygin top, Goryachev system and other systems on the sphere with the cubic, quartic and sextic invariants.…”

“…According to [43] integrable systems on the hyperboloid were not studied in detail, partly because the generalisation from the ellipsoid case seems to be obvious. The main difference is the existence of non-compact trajectories coming from infinity and going away to infinity, which allows us to study scattering on hyperboloids.…”

“…We want to discuss the counterparts of these polynomial invariants on the ellipsoid and hyperboloid. After the construction of integrals of motion, we will be able to proceed to the study of the properties of noncompact trajectories on hyperboloid [43].…”

“…The concept of a Cayley-Klein geometry leads to a unified description and classification of a wide range of the integrable systems. Nevertheless, the modern literature still divides integrable systems on the sphere, ellipsoid or hyperboloid, see [2,4,7,16,17,18,20,21,24,25,38,39,42,43,48] and references within.…”

Integrable billiards on a Minkowski hyperboloid: extremal polynomials and topology

“…For more details we refer to [37]. We also note the recent paper [50] devoted to geodesics on a hyperboloid in Euclidean space.…”