  Thermodynamic fluctuation metrics in Ruppeiner's formalism are worked out for
Kerr-AdS black holes in the extended state space. The implications of
constraints upon the state space geometry and their correspondence with
thermodynamical ensembles are explicitly worked out in the most general
setting. The state space scalar curvature for a given ensemble is found to be
sensitive to the instabilities/phase transitions therein. In particular, it is
found that the appropriate Ruppeiner scalar curvature does encode critical
phenomena in the Kerr-AdS black holes. A detailed study is undertaken of the
curvature contour of the state space of the 4d Kerr-AdS black hole and suitable
inferences are drawn. In particular, thermodynamic geometry suggests an
instability in the Schwarzschild-AdS limit for all the ensembles except the
pressure ensemble which is equivalent to the unextended state space of the
Kerr-AdS black holes. The extrinsic geometry of the ensemble hypersurfaces is
introduced and its relevance to constrained thermodynamic fluctuations
discussed. A new interpretation for the thermodynamic curvature of black hole
systems is suggested.
