Compatibility of a Jacobi structure and a Riemannian structure on a Lie algebroid

TL;DR

This paper generalizes the compatibility between Jacobi and Riemannian structures from manifolds to Lie algebroids, extending known geometric structures like Poisson, contact, and locally conformally symplectic structures.

Contribution

It introduces a framework for compatibility of Jacobi and Riemannian structures on Lie algebroids, broadening the scope of geometric structures studied.

Findings

Extension of compatibility concepts to Lie algebroids

Identification of Riemann-Poisson, (1/2)-Kenmotsu, and locally conformally Kähler structures in this context

Generalization of previous manifold-based results

Abstract

In a preceding paper we introduced a notion of compatibility between a Jacobi structure and a Riemannian structure on a smooth manifold. We proved that in the case of fundamental examples of Jacobi structures : Poisson structures, contact structures and locally conformally symplectic structures, we get respectively Riemann-Poisson structures in the sense of M. Boucetta, (1/2)-Kenmotsu structures and locally conformally K\"ahler structures. In this paper we are generalizing this work to the framework of Lie algebroids.