  We consider the space of Kahler metrics as a Riemannian submanifold of the
space of Riemannian metrics, and study the associated submanifold geometry. In
particular, we show that the intrinsic and extrinsic distance functions are
equivalent. We also determine the metric completion of the space of Kahler
metrics, making contact with recent generalizations of the Calabi-Yau Theorem
due to Dinew, Guedj-Zeriahi, and Kolodziej. As an application, we obtain a new
analytic stability criterion for the existence of a Kahler-Einstein metric on a
Fano manifold in terms of the Ricci flow and the distance function. We also
prove that the Kahler-Ricci flow converges as soon as it converges in the
metric sense.
