三维等半轴椭球上的测地线流
Geodesic flow on three dimensional ellipsoids with equal semi-axes
摘要：
在我们以前对具有相等中间半轴的三维椭球上的测地线流的研究的基础上，本文研究了其余的情形：具有两组相等半轴的椭球具有$SO(2) \times SO(2)$对称性，具有相等的较大或较小半轴的椭球具有SO（2）对称性，以及具有三个半轴的椭球具有SO（3）对称性。所有这些情况都是刘维尔可积的，对称性的减少导致低维椭球上的奇异约化系统。对能量-动量映射及其奇异纤维的临界值进行了完全分类。在SO（2）对称的情况下，存在余秩为1的退化临界点，其余临界点均为非退化临界点。我们证明了在具有$SO(2) \times SO(2)$对称性的情况下，存在三个整体作用量，并且能量-动量映射下的能量面像是凸多面体。SO（3）对称的情形是不可交换可积的，我们证明了能量-Casimir映射的正则点上的纤维是$S^2$上的$T^2$丛.
英文摘要：
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 $S^2$.
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