Connections in Poisson Geometry I: Holonomy and Invariants

TL;DR

This paper explores contravariant connections on Poisson manifolds, introducing Poisson holonomy and invariants to better understand their global properties, building on Vaisman's notion of contravariant derivatives.

Contribution

It introduces the concepts of Poisson holonomy and new invariants for Poisson manifolds using contravariant connections, advancing the understanding of their global structure.

Findings

Defined Poisson holonomy for Poisson manifolds

Introduced new invariants based on contravariant connections

Demonstrated the importance of these tools in global Poisson geometry

Abstract

We discuss contravariant connections on Poisson manifolds. For vector bundles, the corresponding operational notion of a contravariant derivative had been introduced by Izu Vaisman. We show that these connections play an important role in the study of global properties of Poisson manifolds and we use them to define Poisson holonomy and new invariants of Poisson manifolds.