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

![enter image description here][1]


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.




  [1]: https://i.sstatic.net/gdtdH.gif