# Geodesics on Compact Manifolds

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?

Thanks,

S.

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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