In physics, geometrothermodynamics (GTD) is a formalism developed in 2007 by Hernando Quevedo to describe the properties of thermodynamic systems in terms of concepts of differential geometry.
Consider a thermodynamic system in the framework of classical equilibrium thermodynamics. The states of thermodynamic equilibrium are considered as points of an abstract equilibrium space in which a Riemannian metric can be introduced in several ways. In particular, one can introduce Hessian metrics like the Fisher information metric, the Weinhold metric, the Ruppeiner metric and others, whose components are calculated as the Hessian of a particular thermodynamic potential. 
Another possibility is to introduce metrics which are independent of the thermodynamic potential, a property which is shared by all thermodynamic systems in classical thermodynamics. Since a change of thermodynamic potential is equivalent to a Legendre transformation, and Legendre transformations do not act in the equilibrium space, it is necessary to introduce an auxiliary space to correctly handle the Legendre transformations. This is the so-called thermodynamic phase space. If the phase space is equipped with a Legendre invariant Riemannian metric, a smooth map can be introduced that induces a thermodynamic metric in the equilibrium manifold. The thermodynamic metric can then be used with different thermodynamic potentials without changing the geometric properties of the equilibrium manifold. One expects the geometric properties of the equilibrium manifold to be related to the macroscopic physical properties. 
The details of this relation can be summarized in three main points:
Geometric aspects.
The main ingredient of GTD is a (2"n" + 1)-dimensional manifold formula_1 
with coordinates formula_2, where formula_3 is an arbitrary thermodynamic potential, formula_4, formula_5, are the
extensive variables, and formula_6 the intensive variables. It is also
possible to introduce in a canonical manner the fundamental
one-form formula_7 (summation over repeated indices) with formula_8, which satisfies the condition formula_9, where formula_10 is the number of thermodynamic
degrees of freedom of the system, and is invariant with respect to
Legendre transformations
 formula_11
where formula_12 is any disjoint decomposition of the set of indices formula_13,
and formula_14. In particular, for formula_15 and formula_16 we obtain
the total Legendre transformation and the identity, respectively.
It is also assumed that in formula_17 
there exists a metric formula_18 which is also
invariant with respect to Legendre transformations. The triad
formula_19 defines a Riemannian contact manifold which is
called the thermodynamic phase space (phase manifold). The space of
thermodynamic equilibrium states (equilibrium manifold) is an
n-dimensional Riemannian submanifold formula_20
induced by a smooth map formula_21,
i.e. formula_22, with formula_23
and formula_24, such that formula_25 holds, where formula_26 is the
pullback of formula_27. The manifold formula_28 is naturally equipped
with the Riemannian metric formula_29. The purpose of GTD is
to demonstrate that the geometric properties of formula_28 are
related to the thermodynamic properties of a system with fundamental
thermodynamic equation formula_23.
The condition of invariance with respect total Legendre transformations leads to the metrics
 formula_32
 formula_33
where formula_34 is a constant diagonal matrix that can be expressed in terms of formula_35 and
formula_36, and formula_37 is an arbitrary Legendre invariant function of formula_38. The metrics formula_39 and formula_40 have been used to describe thermodynamic systems with first and second order phase transitions, respectively. The most general metric which is invariant with respect to partial Legendre transformations is
 formula_41
The components of the corresponding metric for the equilibrium manifold formula_42 can be computed as
 formula_43
Applications.
GTD has been applied to describe laboratory systems like the ideal gas, van der Waals gas, the Ising model, etc., more exotic systems like black holes in different gravity theories, in the context of relativistic cosmology, and to describe chemical reactions