Geometrothermodynamics

We present the fundamentals of geometrothermodynamics, an approach to studythe properties of thermodynamic systems in terms of differential geometricconcepts. It is based, on the one hand, upon the well-known contact structureof the thermodynamic phase space and, on the other hand, on the metricstructure of the space of thermodynamic equilibrium states. In order to makethese two structures compatible we introduce a Legendre invariant set ofmetrics in the phase space, and demand that their pullback generates metrics onthe space of equilibrium states. We show that Weinhold's metric, which wasintroduced {\it ad hoc}, is not contained within this invariant set. We proposealternative metrics which allow us to redefine the concept of thermodynamiclength in an invariant manner and to study phase transitions in terms ofcurvature singularities.Comment: Revised version, to be published in Jour. Math. Phy