Compatibility between Jacobi structures and pseudo-Riemannian cometrics on Jacobi algebroids

TL;DR

This paper introduces a generalized notion of compatibility between Jacobi structures and pseudo-Riemannian cometrics on Jacobi algebroids, extending known concepts from Poisson geometry and linking to Sasakian structures.

Contribution

It defines compatibility in the Jacobi algebroid context, shows its preservation under Poissonization, and characterizes contact pseudo-metric structures as Sasakian when compatible.

Findings

Compatibility is preserved under Poissonization.

Contact pseudo-metric structures are Sasakian if compatible.

Generalizes Poisson structure compatibility to Jacobi algebroids.

Abstract

We define compatibility between Jacobi structures and pseudo-Riemannian cometrics on Jacobi algebroids. This notion is a generalization of the compatibility between Poisson structures and pseudo-Riemannian cometrics on manifolds, which was defined by Boucetta. We show that the compatibility with a cometric is ``preserved'' by the Poissonization of a Jacobi structure. Furthermore, we prove that for a contact pseudo-metric structure on a manifold, satisfying the compatibility condition is equivalent to being a Sasakian pseudo-metric structure.