On the Riemannian Geometry of tangent Poisson-Lie group

Let $(G,{\mathrm{\Pi }}_G,\tilde{g})$ be a Poisson-Lie group equipped with a left invariant pseudo-Riemannian metric $\tilde{g}$ and let $(TG,{\mathrm{\Pi }}_{TG},{\tilde{g}}^c)$ be the Sanchez de Alvarez tangent Poisson-Lie group of $G$ equipped with the left invariant pseudo-Riemannian metric ${\tilde{g}}^c$, complete lift of $\tilde{g}$. In this paper, we express respectively the Levi-Civita connection, curvature and metacurvature of $(TG,{\mathrm{\Pi }}_{TG},{\tilde{g}}^c)$ in terms of the Levi-Civita connection, curvature and metacurvature of the basis Poisson-Lie group $(G,{\mathrm{\Pi }}_G,\tilde{g})$ and we prove that the space of differential forms ${\mathrm{\Omega }}^{\ast }(G)$ on $G$ is a differential graded Poisson algebra if, and only if, ${\mathrm{\Omega }}^{\ast }(TG)$ is a differential graded Poisson algebra . Moreover, we prove that the triplet $(G,{\mathrm{\Pi }}_G,\tilde{g})$ is a pseudo-Riemannian Poisson-Lie group if, and only if, $(TG,{\mathrm{\Pi }}_{TG},{\tilde{g}}^c)$ is also a pseudo-Riemannian Poisson-Lie group and we give an example of 6-dimensional pseudo-Riemannian Sanchez de Avarez tangent Poisson-Lie group.