Geodesics on path spaces and double category

Let $M$ be a Riemannian manifold and $\mathcal{P}M$ be the space of all smooth paths on $M$. We describe geodesics on path space $\mathcal{P}M$. Normal neighbourhood structure on $\mathcal{P}M$ has been discussed. We identify paths on $M$ under “back-track” equivalence. Under this identification we show that if $M$ is complete, then geodesics on path space yield a double category.We gave a physical interpretation of this double category.