Abstract
Let (G, Lambda(G), <,>(G)) be a Poisson Lie-group equipped with a left invariant Riemannian metric and H a normal and closed coisotropic subgroup of G. In this paper, we give necessary and sufficient conditions so that the compatibility conditions between the Poisson tensor A G and the metric <,>(G) on G remain verified on the reduced Poisson-Lie group (G/H, Lambda(G/H), <,>(G/H)). In particular, if the immersed dual group (G/H)* is totally geodesic in G* then, these conditions are satisfied.