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According to Jairo comment on the first version of this question I revise the question as follwos; 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 $m=n$

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 $m=n$

Let $M$ be a compact real analytic Riemann surface. Is it true to say that:

There are at most a finite number of disjoint simple closed geodesics on $M$.

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 $m=n$

According to Jairo comment on the first version of this question I revise the question as follwos; 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 $m=n$