Cotangent paths as coisotropic subsets for local functions

TL;DR

This paper proves a local function version of a classical Poisson characterization using cotangent paths, avoiding Banach manifold techniques, thus extending the result to cases like cotangent loops.

Contribution

It introduces a new approach using local functions on path spaces to characterize Poisson structures without Banach manifold assumptions.

Findings

Establishes a local function criterion for Poisson bivectors.

Shows cotangent paths form a coisotropic set in the local function framework.

Extends classical results to the periodic case where Banach structures fail.

Abstract

We establish a local function version of a classical result claiming that a bivector field on a manifold $M$ is Poisson if and only if cotangent paths form a coisotropic set of the infinite dimensional symplectic manifold of paths valued in $T^*M$. Our purpose here is to prove this result without using the Banach manifold setting, setting which fails in the periodic case because cotangent loops do not form a Banach sub-manifold. Instead, we use local functions on the path space, a point of view that allows to speak of a coisotropic set.