Extrinsic curvature in geometrothermodynamics

Based on the first law of black hole thermodynamics, we yield the metric of geometrothermodynamics. Choosing a hypersurface of constant extensive variable, the extrinsic curvature scalar is calculated and general correspondence of singularities between the extrinsic curvature scalar and the specific heat is discovered. This correspondence is further shown to exist in the constant intensive variable ensemble. We also show that extrinsic curvature scalar can reflect thermodynamic stability information, in spite of considering fluctuations of different thermodynamic variables or choosing different hypersurfaces.