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



Are compact complete geodesics closed?
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 $M$.


