Geodesics on path spaces and double category

Abstract

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.

20 pages, Added a section explaining physical interpretation of the double category

References in corpus (4)

• Higher Gauge Theory: 2-Connections on 2-Bundles
• Parallel Transport over Path Spaces
• Existence of 'Darboux chart' on loop space
• Path space as manifold, manifold of maps, infinite dimensional manifold