On the symplectic realization of Poisson-Nijenhuis manifolds

TL;DR

This paper investigates the symplectic realization of Poisson-Nijenhuis manifolds, extending techniques from Poisson geometry to establish conditions under which such realizations exist, with illustrative examples.

Contribution

It introduces a new method to construct symplectic realizations of Poisson-Nijenhuis manifolds under specific conditions, expanding the understanding of their geometric structure.

Findings

Existence of a nondegenerate Poisson-Nijenhuis structure near the zero-section

Application of Crainic and Marcut's technique to Poisson-Nijenhuis manifolds

Examples illustrating the symplectic realization process

Abstract

We consider the problem of the symplectic realization of a Poisson-Nijenhuis manifold. By applying a new technique developed by M. Crainic and I. Marcut for the study of the above problem in the case of a Poisson manifold, we establish the existence, under a condition, of a nondegenerate Poisson-Nijenhuis structure on an open neighborhood of the zero-section of the cotangent bundle of the manifold, which symplectizes the initial structure. Additionally, we present some examples.