Geometrothermodynamics

We present the fundamentals of geometrothermodynamics, an approach for studying the properties of thermodynamic systems with $n$ degrees of freedom in terms of differential geometric concepts. It is based, on the one hand, upon the well-known contact structure of the $(2n+1)-$thermodynamic phase space and, on the other hand, on the metric structure of the $n-$dimensional space of thermodynamic equilibrium states. In order to make these two structures compatible we introduce the set of Legendre invariant metrics in the thermodynamic phase space and demand that by using the pullback they generate $n-$dimensional metrics on the space of thermodynamic equilibrium states. We show that Weinhold's metric, which was introduced {\it ad hoc}, is not contained within this invariant set, but we also propose several new Legendre invariant metrics which reproduce the most important geometric properties of Weinhold's metric