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Let $M$ be a compact manifold (hence complete). Let $p$ be any point on $M$. Is it true that we can always find a geodesic loop based at $p$? If $M$ is non-simply connected it is true as each homotopy class can be represented by geodesic loop. But what if $M$ is simply-connected?

Edit : By a geodesic loop $\gamma$ at $p$ I mean for some $t$, $\gamma(0)=\gamma(t)=p$.

Bingo
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  • even in the not simply connected case I doubt that you can prescribe a point through which the (closed) geodesic has to pass. There are some well known theorems which proof the existence of some closed geodesic loops on closed simply connected $M$, I don't know of any result where you can prescribe a point of the geodesic. But that is knowledge from about 1995, maybe today more is known.... – Thomas Dec 15 '14 at 14:55
  • @Thomas: I have edited my post. – Bingo Dec 15 '14 at 15:11
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    (Just in case if you havent found that) http://mathoverflow.net/questions/131175/closed-geodesic-loops-around-points-in-compact-manifolds –  Mar 10 '15 at 07:12

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