We show that every forward complete Finsler manifold of infinite fundamental group and not homotopy-equivalent to $S^1$ has infinitely many geometrically distinct geodesics joining any given pair of points p and q. In the special case in which ${\beta }_1(M;\mathbb{Z})\ge 1$ and M is closed, the number of geometrically distinct geodesics between two points grows at least logarithmically.

中文翻译：

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关于测地弦的增长率

我们证明了无限基本组的每个前向完整Finsler流形并不等于 ${\mathrm{小号}}^{\mathrm{1个}}$具有无限多的几何上不同的测地线，它们连接任何给定的点p和q。在特殊情况下${\beta }_{\mathrm{1个}}（\mathrm{中号};\mathbb{ž}）\ge \mathrm{1个}$当M闭合时，两点之间几何上不同的测地线数至少对数增长。