Are the geometries of the first and second laws of thermodynamics compatible? (Q1946342)

The authors consider two different approaches to differential geometric formulations of phenomenological thermodynamics. One embeds the space of equilibrium states of a system of \(n\) extensive state variables, \(V\), into a contact manifold of \((2n+1)\) dimensions, \(M\), accordingly to the first law of thermodynamics, i.e., the extensive variables \((x_1,x_2,\dots,x_n)=:x\) are mapped to \((x_1,x_2,\dots,x_n,\frac{\partial u}{\partial x_1}(x), \frac{\partial u}{\partial x_2}(x),\dots,\frac{\partial u}{\partial x_n}(x),u(x))\) in some canonical chart, where \(u\) denotes the internal energy and hence \(y_i=\frac{\partial u}{\partial x_i}\) are the intensive variables. The other approach uses the second law of thermodynamics to endow the \(n\)-dimensional space \(V\) with a Riemannian metric given by the Hessian 2-form \(\mathrm{Hess}(u)\) of the convex function \(u\). The authors assume that compatibility of these two approaches requires the embedding of \(V\) into \(M\) to be in a sense isometric. Investigating Riemannian geometry in contact manifolds, the authors use a result of \textit{D. E. Blair} [Riemannian geometry of contact and symplectic manifolds. Boston, MA: Birkhäuser (2002; Zbl 1011.53001)] and prove a theorem stating that the embedding under consideration cannot be isometric. The authors investigate whether there is an isometric embedding of \(V\) into \(M\) at all. To that end, they take into account that in the second approach the Riemannian metric is given by a Hessian two-form and relationships between contact and symplectic as well as between symplectic and Kählerian structures. They construct an isometric embedding but show why it is useless for the present purpose. So, with respect to the proposed assumption the answer to the title question is shown to be in the negative.