From 2-covariant tensor fields to Poisson and Jacobi structures
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.