Let $M$ be a compact manifold (w/o boundary). Suppose that there is no closed geodesics on $M$ of length precisely $C$. I am trying to prove that there is an open cover $\{U_j\}$ of $M$ and $\epsilon >0$ with the following property: if $\gamma$ is a unit speed geodesic in $M$, $C-\epsilon < t < C+\epsilon$, then $\gamma(0)$ and $\gamma(t)$ do not lie in the same $U_j$.
It seems like this would be amenable to a compactness argument of some sort, perhaps using an exponential neighborhood at each point. But, I am not seeing a good way to do it. Any suggestions?