TITLE: Surfaces all of whose geodesics are both closed and simple
QUESTION [8 upvotes]: The Zoll surfaces have the property that all of their geodesics are closed.
If one futher stipulates that all geodesics are also simple, i.e., non-self-intersecting,
does this leave only the sphere?
Apologies for the simplicity of this question, but I am not finding an answer in the literature,
and I suspect many just know this off the top of their head.  Thanks!

REPLY [12 votes]: From Guillemin's "The Radon transform on Zoll surfaces", it follows that there are deformations of $S^2$ which keep all geodesics closed AND simple.