On three-dimensional Poisson quasi-Nijenhuis manifolds and Haantjes structures

TL;DR

This paper characterizes three-dimensional Poisson quasi-Nijenhuis manifolds with non-vanishing Poisson tensors, showing they are deformations of PN structures, involutive, and possess Haantjes structures with Lenard-Magri chains.

Contribution

It provides a characterization and classification of three-dimensional Poisson quasi-Nijenhuis manifolds, linking them to Haantjes structures and Lenard-Magri chains.

Findings

Each such manifold is a deformation of a PN structure.

They are involutive.

They carry a generalized Lenard-Magri chain.

Abstract

In this note we first characterize Poisson quasi-Nijenhuis structures on three-dimensional oriented manifolds whose underlying Poisson tensor never vanishes. We then apply this result to show that each of these structures is (locally) a deformation of a PN structure and is involutive. Finally, we prove that every such three-dimensional Poisson quasi-Nijenhuis manifold is a Haantjes manifold and that it carries a generalized Lenard-Magri chain.