What, if anything, can be said about a surface whose geodesics are all algebraic? One example is of course the sphere.

A class of examples is provided by projective spaces, in which case prime closed geodesics are great circles on spheres. See Klingenberg's Lectures on Closed Geodesics, p. 178, Theorem 5.2.1. The relevant page is not available on Google, but is accessible on Amazon. 

