We study non-reversible Finsler metrics with constant flag curvature 1 on S^2 and show that the geodesic flow of every such metric is conjugate to that of one of Katok's examples, which form a 1-parameter family. In particular, the length of the shortest closed geodesic is a complete invariant of the geodesic flow. We also show, in any dimension, that the geodesic flow of a Finsler metrics with constant positive flag curvature is completely integrable. Finally, we give an example of a Finsler metric on~$S^2$ with positive flag curvature such that no two closed geodesics intersect and show that this is not possible when the metric is reversible or have constant flag curvature

中文翻译：

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$S^2$ 上恒定正旗曲率的 Finsler 度量的测地线行为

我们研究了 S^2 上具有恒定标志曲率 1 的不可逆 Finsler 度量，并表明每个此类度量的测地线流与 Katok 示例之一的测地线流共轭，形成一个 1 参数族。特别地，最短闭合测地线的长度是测地线流的完全不变量。我们还表明，在任何维度上，具有恒定正旗曲率的 Finsler 度量的测地线流是完全可积分的。最后，我们给出了一个在~$S^2$ 上具有正旗曲率的 Finsler 度量的例子，这样没有两个封闭的测地线相交，并表明当度量是可逆的或具有恒定的旗曲率时这是不可能的