TITLE: Number of disjoint simple closed geodesics
QUESTION [10 upvotes]: According to Jairo comment on the first version of this question I revise the question as follows;
Let $g$ be a real analytic Riemannan metric on $S^{2}$. Is it true to say that:
There are at most a finite number of disjoint simple closed geodesics on $S^{2}$.
If the answer is yes put $m$= the sup of the number of such disjoint closed geodesics.
What is a geometric interpretation for this geometric invariant $m$?
For a given $n\in \mathbb{N}$, is there  a real analytic Riemannian metric on $S^{2} $ for which  $m=n$

REPLY [4 votes]: A simpler example seems to be an "accordion surface" (take a sinusoid $y = \sin x,$ rotate around the line $y= 3$) It will have as many parallel simple geodesics as you like.