The authors have recently introduced the class of topological billiards. Topological billiards are glued from elementary planar billiard sheets (bounded by arcs of confocal quadrics) along intervals of their boundaries. It turns out that the integrability of the elementary billiards implies that of the topological billiards. We show that all classical linearly and quadratically integrable geodesic flows on tori and spheres are Liouville equivalent to appropriate topological billiards. Moreover, the linear and quadratic integrals of the geodesic flows reduce to a single canonical linear integral and a single canonical quadratic integral on the billiard. These results are obtained within the framework of the Fomenko–Zieschang theory of the classification of integrable systems.

中文翻译：

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可定向二维表面和拓扑台球上的可积测地线流

作者最近介绍了拓扑台球。拓扑台球是从基本的平面台球板（由共焦二次曲面的弧边界）沿其边界间隔粘在一起的。事实证明，基本台球的可集成性暗示了拓扑台球的可集成性。我们证明了在托里和球面上所有经典的线性和二次可积测地线流量都等于适当的台球形空间。此外，测地流的线性和二次积分在台球上减少为单个规范线性积分和单个规范二次积分。这些结果是在Fomenko-Zieschang可积系统分类理论的框架内获得的。