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Note: I changed the original question to the one, proposed by Robert Bryant in the comment below as it is better formulated.

Assume there is a metric on a manifold with a closed geodesic such that all nearby geodesics are closed too. What can be said about a metric near this geodesic? We do not require that all the geodesics are closed, so the question is local. A good example is a sphere with a deformed metric near the poles, so all the geodesics near the equator are closed.

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would yo be more precise: you mean an open set, in what topological space of curves? – Pietro Majer May 30 '13 at 9:50
@Pietro: I think that the OP means to ask what you can say about a metric such that it has a closed geodesic for which all of the geodesics near that one are closed. For example, if you were to modify the round metric on the sphere near the poles only, then the geodesics near the equator would continue to all be closed. The OP is asking what one can say about metrics that are 'Zoll near a given geodesic'. – Robert Bryant May 30 '13 at 10:40
Robert, precisely. Thank you for reformulating the question as I was thinking how to make it better formulated. – Axel May 30 '13 at 12:46
Even if your metric is Zoll and all geodesics are closed it is very hard to say something about the metric that is not "dynamic" (e.g. behavior of jacobi fields and things like that). Do you have a more precise idea of what you want? – alvarezpaiva May 30 '13 at 14:49
@Axel: there is little that is intuitive about Zoll surfaces. If you allow Finsler metrics you can do really strange things. For example, you can make the polar circles flat (and Riemannian) and still have a Finsler metric on $S^2$ all of whose geodesics are great circles. Even in the Riemannian case it is not clear that you cannot construct a Zoll metric on the two sphere for which the north and south poles (in fact, the whole polar circles) are flat. If I had to bet, I would bet it can be done. There are very many Zoll metrics! – alvarezpaiva May 30 '13 at 18:49

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