Generalized Jacobi--Chasles theorem in non-Euclidean spaces

G. V. Belozerov ; A. T. Fomenko

Sbornik. Mathematics, Tome 215 (2024) no. 9, pp. 1159-1181

Résumé

The classical Jacobi–Chasles theorem states that tangent lines to a geodesic curve on an $n$-axial ellipsoid in $n$-dimensional Euclidean space are also tangent, along with this ellipsoid, to $n-2$ quadrics confocal with it, which are the same for all points on this geodesic. This result ensures the integrability of the geodesic flow on the ellipsoid. As recent results due to Belozerov and Kibkalo show, a similar theorem also holds for an arbitrary intersection of confocal quadrics in Euclidean space. In the present paper it is shown that the geodesic flow on an intersection of several confocal quadrics in a pseudo-Euclidean space $\mathbb R^{p,q}$ or on a constant curvature space is integrable. As a consequence, a similar result is established for confocal billiards on such intersections. It is also shown that in codimension 2 the last result cannot be extended to surfaces not locally isometric to a space of constant curvature. Bibliography: 15 titles.

Détail

Keywords: geodesic flow, integrable system, elliptic coordinates, Jacobi–Chasles theorem.
Mots-clés : confocal quadrics