Geodesic flow on three dimensional ellipsoids with equal semi-axes

TL;DR

This paper classifies the integrable geodesic flows on various three-dimensional ellipsoids with different symmetries, analyzing their energy-momentum maps and singular fibers.

Contribution

It extends previous work by analyzing remaining symmetric cases, classifies critical points, and describes the topology of fibers for geodesic flows on ellipsoids.

Findings

All cases are Liouville-integrable.

The energy-momentum map images are convex polyhedra in certain cases.

Fibers over regular points are tori or torus bundles, depending on symmetry.

Abstract

Following on from our previous study of the geodesic flow on three dimensional ellipsoid with equal middle semi-axes, here we study the remaining cases: Ellipsoids with two sets of equal semi-axes with $SO(2) \times SO(2)$ symmetry, ellipsoids with equal larger or smaller semi-axes with SO(2) symmetry, and ellipsoids with three semi-axes coinciding with SO(3) symmetry. All of these cases are Liouville-integrable, and reduction of the symmetry leads to singular reduced systems on lower-dimensional ellipsoids. The critical values of the energy-momentum maps and their singular fibers are completely classified. In the cases with SO(2) symmetry there are corank 1 degenerate critical points; all other critical points are non-degenreate. We show that in the case with $SO(2) \times SO(2)$ symmetry three global action variables exist and the image of the energy surface under the energy-momentum map is a convex polyhedron. The case with SO(3) symmetry is non-commutatively integrable, and we show that the fibers over regular points of the energy-casimir map are $T^2$ bundles over $S^2$.