Abstract

We show that for a generic Riemannian or reversible Finsler metric on a compact differentiable manifold M of dimension at least three all closed geodesics are simple and do not intersect each other. Using results by Contreras (Ann Math 2(172):761–808, 2010; in: Proceedings of International Congress Mathematicians (ICM 2010) Hyderabad, India, pp 1729–1739, 2011) this shows that for a generic Riemannian metric on a compact and simply-connected manifold all closed geodesics are simple and the number N(t) of geometrically distinct closed geodesics of length \(\le t\) grows exponentially.

中文翻译：

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尺寸为 $$\ge 3$$ 的简单闭合测地线

摘要

我们证明，对于维度至少为三的紧可微流形M上的通用黎曼或可逆芬斯勒度量，所有闭测地线都是简单的并且彼此不相交。使用 Contreras 的结果（Ann Math 2(172):761–808, 2010；in: Proceedings of International Conference Mathematicians (ICM 2010) Hyderabad, India, pp 1729–1739, 2011）这表明，对于紧且简连通流形 所有闭合测地线都是简单的，长度为\(\le t\)的几何上不同的闭合测地线的数量N ( t )呈指数增长。