On Poisson Structure and Curvature

TL;DR

This paper explores the relationship between Poisson structures and curvature in curved space-times, establishing a connection between the algebraic Poisson structure and the geometric Riemann tensor.

Contribution

It introduces a specific relation linking Poisson structures to the Riemann tensor in curved space-times, advancing understanding of noncommutative geometry's classical limit.

Findings

Derived a relation between Poisson structure and Riemann tensor

Illustrated the connection with a simple example

Enhanced understanding of noncommutative geometry in curved space-time

Abstract

We consider a curved space-time whose algebra of functions is the commutative limit of a noncommutative algebra and which has therefore an induced Poisson structure. In a simple example we determine a relation between this structure and the Riemann tensor.