  Let (X,\omega) be a compact K\"ahler manifold. As discovered in the late
1980s by Mabuchi, the set H_0 of K\"ahler forms cohomologous to \omega has the
natural structure of an infinite dimensional Riemannian manifold. We address
the question whether any two points in H_0 can be connected by a smooth
geodesic, and show that the answer, in general, is "no".
