Compatibility between Jacobi structures and pseudo-Riemannian cometrics on 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.