Abstract
Let [Formula: see text] be a Riemannian manifold and [Formula: see text] be the space of all smooth paths on [Formula: see text]. We describe geodesics on path space [Formula: see text]. Normal neighborhoods on [Formula: see text] have been discussed. We identify paths on [Formula: see text] under “back-track” equivalence. Under this identification, we show that if [Formula: see text] is complete, then geodesics on the path space yield a double category. This double category has a natural interpretation in terms of the worldsheets generated by freely moving (without any external force) strings.
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