  We give a notion of compatibility between a Riemannian metric and a Jacobi
structure. We prove that in case of Poisson structures, contact structures and
locally conformally symplectic structures, fundamental examples of Jacobi
structures, we get respectively Riemann-Poisson structures in the sense of M.
Boucetta, $\frac{1}{ 2}$-Kenmotsu structures and locally conformally K\"ahler
structures.
