# Differential operators and contravariant derivatives in Poisson geometry

## TL;DR

This paper explores the relationships between curl operators, Poisson coboundary operators, and contravariant derivatives on Poisson manifolds, introducing the modular operator and providing explicit descriptions in symplectic cases.

## Contribution

It introduces the notion of the modular operator in Poisson geometry and describes differential operators in terms of Poisson connections with vanishing torsion.

## Key findings

- Explicit local descriptions of differential operators in terms of Poisson connections.
- Introduction of the modular operator for oriented Poisson manifolds.
- Explicit description of the modular operator in symplectic manifolds using curvature.

## Abstract

We inquire into the relation between the curl operators, the Poisson coboundary operators and contravariant derivatives on Poisson manifolds to study the theory of differential operators in Poisson geometry. Given an oriented Poisson manifold, we describe locally those two differential operators in terms of Poisson connection whose torsion is vanishing. Moreover, we introduce the notion of the modular operator for an oriented Poisson manifold. For a symplectic manifold, we describe explicitly the modular operator in terms of the curvature 2-section of Poisson connection, analogously to the Weitzenb$\ddot{\rm o}$ck formula in Riemannian geometry.