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Let $(M,g)$ be a compact complete Riemannian manifold and $l$ be infimum of lengths of closed geodesics in $M$, suppose the maximal of the sectional curvature of $M$ is $K$, could we get a lower bound of $l$ in terms of K?

I think maybe one could apply Rauch's comparison theorem to get such kind of bound but I am not sure how to use that. Thank you so much.

San John
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    No. Flat torus. That’s a good example to check whenever trying to do something under the assumption of bounded sectional curvature. – Deane Sep 28 '21 at 23:54
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    I just wanted to add that Deane's example can be promoted to one which works if $M>0$. Namely, consider $S^2\times S^1$ (Riemannian product) where both $S^2$ and $S^1$ have a tiny radius. You can even generalize this to have a large diameter by looking at $S^2\times S^1\times S^1$ with the last $S^1$ really big. – Jason DeVito - on hiatus Sep 29 '21 at 00:46
  • @Deane@Jason DeVito Thank you so much for your comments. – San John Sep 30 '21 at 00:53

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