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note: I find this question In stackexchange math, I would be interest to know how I could be answer this kind of question,I pasted it here as I see it appropriate For MO.

check this link: https://math.stackexchange.com/q/1030616/156150.

Let $(M,g)$ be a compact Riemannian manifold.

Is there an example of a geodesic $c:\mathbb{R}\to M$ s.t. $c(\mathbb{R})$ is compact, $c$ is NOT periodic (i.e. be NOT a closed geodesic) ?

I would be interest for any replies or any comments .Thank you

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    $\begingroup$ Please link to the original question. $\endgroup$ Nov 23, 2014 at 22:44
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    $\begingroup$ The question on math.stackexchange already has an answer. Why did you want to repeat the question here? $\endgroup$ Nov 24, 2014 at 20:00
  • $\begingroup$ this question havn't answer for 3 days before , and i pasted it here as i see it apprpriate for MO $\endgroup$ Nov 24, 2014 at 20:19
  • $\begingroup$ This question appears to be off-topic because it was asked and answered elsewhere $\endgroup$
    – Yemon Choi
    Nov 24, 2014 at 22:34

1 Answer 1

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Let $c:\mathbb R\to M$ be a unit speed geodesic. Then $c':\mathbb R\to UM:=\lbrace X\in TM: \|X\|=1\rbrace\subset TM$ is a flow line for the flow of the geodesic spray $S$. Since $UM$ is compact, flow-lines of $S$ are either periodic or non-compact. Thus the same is true for geodesics on $M$.

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