“According to Gibbs, the geometrical structure of thermodynamics is described by a contact manifold, equipped with the contact form, whose zeros define the laws of thermodynamics:

dε = tdη - pdv

where ε is the energy, t the temperature, η the entropy, p the pressure, and v the volume. Let us call this contact 5-manifold the Gibbs manifold. Gibbs thesis: substances are Legendre submanifolds of the Gibbs manifold.”

“Every mathematician knows it is impossible to understand an elementary course in thermodynamics. The reason is that thermodynamics is based—as Gibbs has explicitly proclaimed—on a rather complicated mathematical theory, on the contact geometry. Contact geometry is one of the few ‘simple geometries’ of the so-called Cartan’s list, but it is still mostly unknown to the physicist—unlike the Riemannian geometry and the symplectic or Poisson geometries, whose fundamental role in physics is today generally accepted.”