In this work we present a brief introduction to the concepts of classical global and local thermodynamic stability from thermodynamic information geometry perspective. We state the necessary and sufficient conditions for stable global thermodynamic equilibrium. Novel relations between Hessians, defined on different thermodynamic manifolds, have been derived, which lead to new expressions for the Weinhold and Ruppeiner thermodynamic metrics. We show that our approach is equivalent to the Nambu bracket formalism traditionally used in the calculation of the heat capacities of the system. Finally, we make our remarks on the concept of local thermodynamic stability and its relation to the global one.