Geometric structures on loop and path spaces

TL;DR

This paper investigates the geometric structures on loop and path spaces, showing the quasi-symplectic structure's independence from the underlying metric and exploring conditions for contact structures.

Contribution

It demonstrates that the quasi-symplectic structure on loop spaces is likely independent of the Riemannian metric and studies conditions for contact structures.

Findings

Quasi-symplectic structure is not dependent on the Riemannian metric.

Conditions for contact structures on loop and path spaces are identified.

The structure's independence suggests limitations in recovering the metric from the symplectic form.

Abstract

Is is known that the loop space associated to a Riemannian manifold admits a quasi-symplectic structure. This article shows that this structure is not likely to recover the underlying Riemannian metric by proving a result that is a strong indication of the "almost" independence of the quasi-symplectic structure with respect to the metric. Finally conditions to have contact structures on these spaces are studied.