From 2-covariant tensor fields to Poisson and Jacobi structures
Abstract
Poisson and Jacobi structures play a fundamental role in the geometric description of many systems arising in classical mechanics. In most cases, the corresponding bivector field is induced by a non-degenerate 2-covariant tensor field. In this paper, we present a unified framework for constructing the associated brackets by studying the Poisson and Jacobi structures induced by these tensors. More specifically, under suitable assumptions on the tensor, we derive a formula for computing the Schouten--Nijenhuis bracket of the associated bivector field in terms of the curvature of a certain distribution and the exterior derivative of a differential form. This formula provides the obstruction to the existence of a Poisson or Jacobi structure. To illustrate the theory, we recover the classical brackets associated with symplectic, locally conformally symplectic, contact, cosymplectic, and cocontact geometries. Finally, we characterise the conditions under which fat bundles and almost cosymplectic structures of order p determine a Jacobi bracket.
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BibTeX
@misc{leon2026from,
title = {{From 2-covariant tensor fields to Poisson and Jacobi structures}},
author = {Manuel de León and Xavier Gràcia and Rubén Izquierdo-López and Ángel Martínez-Muñoz and Xavier Rivas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.20030},
eprint = {2606.20030},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.20030},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - From 2-covariant tensor fields to Poisson and Jacobi structures
AU - Manuel de León
AU - Xavier Gràcia
AU - Rubén Izquierdo-López
AU - Ángel Martínez-Muñoz
AU - Xavier Rivas
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2606.20030
UR - https://arxiv.org/abs/2606.20030
AB - Poisson and Jacobi structures play a fundamental role in the geometric description of many systems arising in classical mechanics. In most cases, the corresponding bivector field is induced by a non-degenerate 2-covariant tensor field. In this paper, we present a unified framework for constructing the associated brackets by studying the Poisson and Jacobi structures induced by these tensors. More specifically, under suitable assumptions on the tensor, we derive a formula for computing the Schouten--Nijenhuis bracket of the associated bivector field in terms of the curvature of a certain distribution and the exterior derivative of a differential form. This formula provides the obstruction to the existence of a Poisson or Jacobi structure. To illustrate the theory, we recover the classical brackets associated with symplectic, locally conformally symplectic, contact, cosymplectic, and cocontact geometries. Finally, we characterise the conditions under which fat bundles and almost cosymplectic structures of order p determine a Jacobi bracket.
ER -