  The Hessian of the entropy function can be thought of as a metric tensor on
the 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 this geometry for some families of black holes. It is flat
for the BTZ and Reissner-Nordstrom black holes, while curvature singularities
occur for the Reissner-Nordstrom-anti-de Sitter and Kerr black holes.
