Homogeneity: From Caratheodory's approach to Gibbs thermodynamics

We consider thermodynamics of homogeneous systems in the framework of Caratheodory's approach. We work in a configuration space where independent variables are all extensive, and we show that the requirement for homogeneity symmetry of the integrable 1-form δQrev, which represents the infinitesimal heat exchanged reversibly by a thermodynamic system, allows to find immediately and explicitly an integrating factor by means of elementary tools of differential geometry. The foliation of the thermodynamic manifold is associated in a natural way with an extensive potential, which is then identified with the metrical entropy S. Both S and the absolute temperature T are derived quantities in this framework. The entropy S corresponds to the fundamental relation in the entropy representation according to Gibbs' approach to thermodynamics, thus an explicit connection between Caratheodory's approach and Gibbs' approach to thermodynamics is obtained. The relation with statistical mechanical formalism is also pointed out.