  In this work we study the intrinsic geometry of the space of Kahler metrics
under various Riemannian metrics. The first part is on the Dirichlet metric. We
motivate its study, we compute its curvature, and we make links with the Calabi
metric, the K-energy, the degenerate complex Hessian equation. The second part
is on the weighted metrics, for which we investigate as well their geometric
properties.
