Answer by Anton Petrunin for Surfaces all of whose geodesics are both closed...
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.
View ArticleSurfaces all of whose geodesics are both closed and simple
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...
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