  The Hessian of the entropy function can be thought of as a metric tensor on
state space. In the context of thermodynamical fluctuation theory Ruppeiner has
argued that the Riemannian geometry of this metric gives insight into the
underlying statistical mechanical system; the claim is supported by numerous
examples. We study these geometries for some families of black holes and find
that the Ruppeiner geometry is flat for Reissner--Nordstr\"om black holes in
any dimension, while curvature singularities occur for the Kerr black holes.
Kerr black holes have instead flat Weinhold curvature.
