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 $t$, $\gamma(0)=\gamma(t)=p$.