Return to Answer

As it was shown by Igor, there is no univeral bound on number of such geodesics. Let me show that the number can not be infinite.

Assume it is possible to get infinite number of such geodesics, say $\gamma_n$, $n\in\mathbb N$. Note that the geodesics $\gamma_i$ for $i\le n$ cut $\mathbb S^2$ into surfaces with geodesic boundaries. By Gauss--Bonnet formula most of these surfaces are cylinders.

By passing to a subsequence, we can assume that for each $n$, the geodesics $\gamma_i$ for $i\le n$ cut $\mathbb S^2$ into two discs and cylinders between $\gamma_i$ and $\gamma_{i+1}$.

Denote by $\gamma_\infty$ the limit of $\gamma_n$ as $n\to\infty$. Note that this limit is defined and the limit geodesic $\gamma_\infty$ is stable.

Given a point $p$ near $\gamma_\infty$ denote by $\ell(p)$ the length of mimimal geodesic loop based at $p$ which goes sufficiently close to $\gamma_\infty$. Note that $\ell$ is an analytic function and its derivatives vanish on $\gamma_\infty$. It follows that $\ell\equiv 0$; i.e. $\gamma_\infty$ lies in a one parameter family of closed geodesics which sweep a neighborhood of $\gamma_\infty$.

Pass to the analytical extension of this one parameter family, lets denote it by $\xi_\tau$. Note that the geodesics in the family stay simple and disjoint locally. Globally, it only may happen that $\xi_0=\xi_c$ for some parameter $c\ne0$. Moreover, since the surface is compact it actually happens  . In this case the surface is a torus, a contradiction.

As it was shown by Igor, there is no univeral bound on number of such geodesics. Let me show that the number can not be infinite.

Assume it is possible to get infinite number of such geodesics, say $\gamma_n$, $n\in\mathbb N$. Note that the geodesics $\gamma_i$ for $i\le n$ cut $\mathbb S^2$ into surfaces with geodesic boundaries. By Gauss--Bonnet formula most of these surfaces are cylinders.

By passing to a subsequence, we can assume that for each $n$, the geodesics $\gamma_i$ for $i\le n$ cut $\mathbb S^2$ into two discs and cylinders between $\gamma_i$ and $\gamma_{i+1}$.

Denote by $\gamma_\infty$ the limit of $\gamma_n$ as $n\to\infty$. Note that this limit is defined and the limit geodesic $\gamma_\infty$ is stable.

Given a point $p$ near $\gamma_\infty$ denote by $\ell(p)$ the length of mimimal geodesic loop based at $p$ which goes sufficiently close to $\gamma_\infty$. Note that $\ell$ is an analytic function and its derivatives vanish on $\gamma_\infty$. It follows that $\ell\equiv 0$; i.e. $\gamma_\infty$ lies in a one parameter family of closed geodesics which sweep a neighborhood of $\gamma_\infty$.

Pass to the analytical extension of this one parameter family, lets denote it by $\xi_\tau$. Note that the geodesics in the family stay simple and disjoint locally. Globally, it only may happen that $\xi_0=\xi_c$ for some parameter $c\ne0$. Moreover, since the surface is compact it actually happens. In this case the surface is a total space of a circle bundle, a contradiction.

As it was shown by Igor, there is no univeral bound on number of such geodesics. Let me show that the number can not be infinite.

Assume it is possible to get infinite number of such geodesics.

From analyticity, we get one-parameter family of such geodesics and we can pass to its analytic extension $\gamma_\tau$.

Note that the geodesics in the family stay simple and disjoint locally. Globaly it only may happen that $\gamma_0=\gamma_c$ for some parameter $c>0$. In this case the surface is a torus.

As it was shown by Igor, there is no univeral bound on number of such geodesics. Let me show that the number can not be infinite.

Assume it is possible to get infinite number of such geodesics, say $\gamma_n$, $n\in\mathbb N$. Note that the geodesics $\gamma_i$ for $i\le n$ cut $\mathbb S^2$ into surfaces with geodesic boundaries. By Gauss--Bonnet formula most of these surfaces are cylinders.

By passing to a subsequence, we can assume that for each $n$, the geodesics $\gamma_i$ for $i\le n$ cut $\mathbb S^2$ into two discs and cylinders between $\gamma_i$ and $\gamma_{i+1}$.

Denote by $\gamma_\infty$ the limit of $\gamma_n$ as $n\to\infty$. Note that this limit is defined and the limit geodesic $\gamma_\infty$ is stable.

Given a point $p$ near $\gamma_\infty$ denote by $\ell(p)$ the length of mimimal geodesic loop based at $p$ which goes sufficiently close to $\gamma_\infty$. Note that $\ell$ is an analytic function and its derivatives vanish on $\gamma_\infty$. It follows that $\ell\equiv 0$; i.e. $\gamma_\infty$ lies in a one parameter family of closed geodesics which sweep a neighborhood of $\gamma_\infty$.

Pass to the analytical extension of this one parameter family, lets denote it by $\xi_\tau$. Note that the geodesics in the family stay simple and disjoint locally. Globally, it only may happen that $\xi_0=\xi_c$ for some parameter $c\ne0$. Moreover, since the surface is compact it actually happens . In this case the surface is a torus, a contradiction.

Consider the surface which look like a bone, you may think it is embedded into $\mathbb R^3$ and you can make it analytic.

To be more precise, take surface of genus 2 with constant curvature $-1$ and two very short geodesics; cut it along the geodesics, put spherical caps instead and smooth to make it analytic.

This bone has many closed simple geodesics, imagine a thread which goes around one of the "horns" on the left side, then both ends go parallel $n$ times around the middle part and then meet on the other side of a horn on the right side.

As it was shown by Igor, there is no univeral bound on number of such geodesics. Let me show that the number can not be infinite.

Assume it is possible to get infinite number of such geodesics.

From analyticity, we get one-parameter family of such geodesics and we can pass to its analytic extension $\gamma_\tau$.

Note that the geodesics in the family stay simple and disjoint locally. Globaly it only may happen that $\gamma_0=\gamma_c$ for some parameter $c>0$. In this case the surface is a torus.