Path space as manifold, manifold of maps, infinite dimensional manifold

TL;DR

This paper demonstrates that the space of all smooth paths on an n-dimensional manifold forms an infinite-dimensional smooth manifold modeled over a complete normable space, exploring its geometric structures and relations.

Contribution

It establishes the smooth manifold structure of path spaces and investigates their geometric properties and connections to the ambient manifold.

Findings

Path space $PM$ is a smooth manifold modeled over a complete normable space.

Various geometric structures on path spaces are characterized.

Relations between path space geometry and the ambient manifold are analyzed.

Abstract

Let $M$ be any $n$ dimensional smooth manifold and $PM$ be the space of all smooth paths, then we showed that $PM$ is a smooth manifold modelled over a complete normable space. We discussed many geometric structure on Path spaces and its relation to ambient space.