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Let $M$ be a compact Riemmanian manifold. Let $G$ denote the set of all geodesics of $M$. If $\gamma\in G$ let $l(\gamma)$ denote its length. Let $S=\sup\{l(\gamma): \gamma\in G\}$

Suppose $S<\infty$. How can we estimate $S$ geometrically?

Edit: I changed some assumptions.

Thanks

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    @Tomás: I don't really know anything about it. Besse has a book called "Manifolds all of whose geodesics are closed" which might contain the kind of information you're looking for. Here's a single data point: In a circle of intrinsic diameter $\pi r$, all geodesics are closed for stupid reasons and $S = \pi r$. On the otherhand, we can embed the circle isometrically into $\mathbb{R}^3$ as a tightly wound coil, so that the extrinsic diameter can be made as small as we like. This means that the diameter of the image of an embedding tells you nothing about $S$.2012-11-02

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