Thermodynamic geometry of charged rotating BTZ black holes

We study the thermodynamics and the thermodynamic geometries of charged rotating Banados-Teitelboim-Zanelli black holes in (2 + 1)-gravity. We investigate the thermodynamics of these systems within the context of the Weinhold and Ruppeiner thermodynamic geometries and the recently developed formalism of geometrothermodynamics. Considering the behavior of the heat capacity and the Hawking temperature, we show that Weinhold and Ruppeiner geometries cannot describe completely the thermodynamics of these black holes and of their limiting case of vanishing electric charge. In contrast, the Legendre invariance imposed on the metric in geometrothermodynamics allows one to describe the charged rotating Banados-Teitelboim-Zanelli black holes and their limiting cases in a consistent and invariant manner.