On the structure of homogeneous local Poisson brackets

TL;DR

This paper investigates the structure of homogeneous local Poisson brackets of degree k on loop spaces, demonstrating that certain constructed connections are flat, which advances understanding of their geometric properties.

Contribution

It introduces explicit linear combinations of standard connections associated with these brackets and proves their flatness, revealing new geometric insights.

Findings

k connections are flat

Explicit linear combinations define these flat connections

Advances understanding of Poisson bracket geometry

Abstract

We consider an arbitrary Dubrovin-Novikov bracket of degree $k$, namely a homogeneous degree $k$ local Poisson bracket on the loop space of a smooth manifold $M$ of dimension $n$, and show that $k$ connections, defined by explicit linear combinations with constant coefficients of the standard connections associated with the Poisson bracket, are flat.