  We classify compact surfaces with torsion-free affine connections for which
every geodesic is a simple closed curve. In the process, we obtain completely
new proofs of all the major results concerning the Riemannian case.
  In contrast to previous work, our approach is twistor-theoretic, and depends
fundamentally on the fact that, up to biholomorphism, there is only one complex
structure on CP2.
