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Let $M$ be a compact, connected smooth manifold. If $p, q$ are points in $M$, is there always a geodesic that goes from $p$ to $q$?

I know that this is certainly not true if $M$ is not compact, but I couldn't find a counterexample for the compact case.

Can anybody help me out?



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Notice that your manifold has to have a metric for the question to make sense (you could get by with a projective class of connections on $M$...) – Mariano Suárez-Alvarez Nov 10 '10 at 5:24
Indeed, I should have written "Riemannian manifold", I guess. – Sam Nov 10 '10 at 6:01
up vote 10 down vote accepted

Yes. This is part of the classical Hopf-Rinow theorem, q.v.

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Thanks Mariano. I don't know Hopf-Rinow yet - I'm still about 70 pages away from it! =) – Sam Nov 10 '10 at 6:00

Just minimize the energy of the path in the homotopy class of paths connecting p and q.

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What's the missing step here? Presumably there's some fact about the space of paths from p to q in a compact manifold that doesn't hold in general that I just don't know about. – Aaron Mazel-Gee Dec 8 '10 at 5:25
If the manifold is not complete, then you have a problem. Take for instance R^2 with the origin removed, p=(-1,0) and q=(1,0). Then there is no geodesic between them. On the other hand, if the Riemannian metric makes the manifold complete, you've won! – OrbiculaR Dec 8 '10 at 15:39

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