Fundamentals of Geometrothermodynamics

We present the basic mathematical elements of geometrothermodynamics which is a formalism developed to describe in an invariant way the thermodynamic properties of a given thermodynamic system in terms of geometric structures. First, in order to represent the first law of thermodynamics and the general Legendre transformations in an invariant way, we define the phase manifold as a Legendre invariant Riemannian manifold with a contact structure. The equilibrium manifold is defined by using a harmonic map which includes the specification of the fundamental equation of the thermodynamic system. Quasi-static thermodynamic processes are shown to correspond to geodesics of the equilibrium manifold which preserve the laws of thermodynamics. We study in detail the equilibrium manifold of the ideal gas and the van der Waals gas as concrete examples of the application of geometrothermodynamics.