The Smooth Structure of the Space of Piecewise-Smooth Loops

TL;DR

This paper investigates the manifold structure of specific spaces of piecewise-smooth loops in finite-dimensional manifolds, analyzing the diffeomorphism group's action and proposing a completion to bounded variation loops for improvement.

Contribution

It establishes a smooth manifold structure for certain piecewise-smooth loops and explores the limitations of the circle's diffeomorphism group action on these spaces.

Findings

Successfully defined manifold structure for particular piecewise-smooth loops

Identified limitations of the circle diffeomorphism group's action

Proposed completion to bounded variation loops to address action issues

Abstract

We consider the problem of defining the structure of a smooth manifold on the various spaces of piecewise-smooth loops in a smooth finite dimensional manifold. We succeed for a particular type of piecewise-smooth loops. We also examine the action of the diffeomorphism group of the circle. It is not a useful action on the manifold that we define. We consider how one might fix this problem and conclude that it can only be done by completing to the space of loops of bounded variation.