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We study the geodesic equation for the Dirichlet (gradient) metric in thespace of Kaehler potentials. We first solve the initial value problem for thegeodesic equation of the combination metric, including the gradient metric. Wethen discuss a comparison theorem between it and the Calabi metric.  As geometric motivation of the combination metric, we find that the Ebinmetric restricted to the space of type II deformations of a Sasakian structureis the sum of the Calabi metric and the gradient metric.

On the geodesic problem for the Dirichlet metric and the Ebin metric on the space of Sasakian metrics