This part comes from Petersen's book in the chapter of Hopf–Rinow. I don't quite understand how compactness is relevant here.
Isn't this just a sole application of the theorem? What does compactness of $\hat{K}$ have to do with this?
2026-03-27 18:08:21.1774634901
Geodesics in compact subsets of $TM$?
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The point is you can get an $\epsilon>0$ which works for every point in $\hat{K}$ simultaneously. The Existence theorem only gives the existence of an $\epsilon>0$ locally, you will need a way to get something global. If there are finitely many neighborhoods and epsilons this is not a problem, take the minimum.