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H. Saito
H. Saito
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Let $M$ be a closed Riemannian manifold of dimension $n \geq 2$. I am looking at sufficient conditions for the following two properties:

  1. existence of closed geodesics of arbitrarily long length on $M$. Of course, we assume that a closed geodesic must be traversed only once to find its length

  2. existence of arbitrarily long geodesic segments. By geodesic segment, we mean geodesics with domains restricted to closed intervals, and whose start and endpoints are not necessarily the same.

One can see that on round spheres, neither of the above is possible. On the other hand, on negatively curved manifolds, both are true (due to ergodicity of the geodesic flow, I presume). They are true also on tori. Also, the first condition implies the second condition.

Being a beginner, I am not sure what is known in the literature, or how to go about thinking about such questions. Any advice is highly appreciated!

Let $M$ be a closed Riemannian manifold of dimension $n \geq 2$. I am looking at sufficient conditions for the following two properties:

  1. existence of closed geodesics of arbitrarily long length on $M$. Of course, we assume that a closed geodesic must be traversed only once to find its length

  2. existence of arbitrarily long geodesic segments. By geodesic segment, we mean geodesics with domains restricted to closed intervals, and whose start and endpoints are not necessarily the same.

One can see that on round spheres, neither of the above is possible. On the other hand, on negatively curved manifolds, both are true (due to ergodicity of the geodesic flow, I presume). They are true also on tori. Being a beginner, I am not sure what is known in the literature, or how to go about thinking about such questions. Any advice is highly appreciated!

Let $M$ be a closed Riemannian manifold of dimension $n \geq 2$. I am looking at sufficient conditions for the following two properties:

  1. existence of closed geodesics of arbitrarily long length on $M$. Of course, we assume that a closed geodesic must be traversed only once to find its length

  2. existence of arbitrarily long geodesic segments. By geodesic segment, we mean geodesics with domains restricted to closed intervals, and whose start and endpoints are not necessarily the same.

One can see that on round spheres, neither of the above is possible. On the other hand, on negatively curved manifolds, both are true (due to ergodicity of the geodesic flow, I presume). They are true also on tori. Also, the first condition implies the second condition.

Being a beginner, I am not sure what is known in the literature, or how to go about thinking about such questions. Any advice is highly appreciated!

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H. Saito
H. Saito
  • 31
  • 2

Lengths of closed geodesics and geodesic segments

Let $M$ be a closed Riemannian manifold of dimension $n \geq 2$. I am looking at sufficient conditions for the following two properties:

  1. existence of closed geodesics of arbitrarily long length on $M$. Of course, we assume that a closed geodesic must be traversed only once to find its length

  2. existence of arbitrarily long geodesic segments. By geodesic segment, we mean geodesics with domains restricted to closed intervals, and whose start and endpoints are not necessarily the same.

One can see that on round spheres, neither of the above is possible. On the other hand, on negatively curved manifolds, both are true (due to ergodicity of the geodesic flow, I presume). They are true also on tori. Being a beginner, I am not sure what is known in the literature, or how to go about thinking about such questions. Any advice is highly appreciated!