One of the most striking features of thermodynamics is the division into two
sets of those variables that appear in thermodynamic constraints: there are
*intensities* (like Pressure and Temperature, E and B) which appear in the
thermodynamic functions as variables homogeneous of degree 0, and there are
*extensive quantities* (like Volume, Energy and Entropy, D and H) which
appear as variables homogeneous of degree 1. Classically, there is no way to
distinguish (geometrically) these two different species. However, it has long
been recognized in the calculus of variations of a singly parameterized
function, L(x,V;t), with dx-Vdt=0 as a constraint, that homogeneous functions
play a critical role. The variational integral is independent of the choice of
parameterization, t, if and only if the function L(x,V,t) is homogeneous of
degree one in the variables, V. This means that V, or anything proportional to
V, such as U=b(x,V) V can be used in the constraints, dx-Vdt = 0 = dx-Uds,
where the new parameterization function satisfies ds - bdt = 0. Then the
variational procedure on the variational integrand L(x,V,t)dt or on the
variational integrand L(x,U,s)ds will give the same minimal result.

When a vector field, V, is considered to be the same vector field, independent
from any factor of scaling (or renormalization), then Chern calls the vector
field, V, a *projectivized* vector field. A projective geometry is a
geometry that is independent from scales, and can be modeled as the set of all
rays through some point of perspective, named the origin. The mathematical
vehicle of parametric independent variational calculus distinguishes functions
which are homogeneous of degree 1. Such is the arena of thermodynamics, and
therefor projective geometries share some of the properties of an (equilibrium)
thermodynamic system.

But there is more to the story, for in
1919 Finsler developed a thesis in which
he studied a geometry which, if unconstrained, was not only non-euclidean, but
also non-Riemannian. Special constraints would reduce Finsler geometries to the
more common Riemannian geometries. The new feature permitted in Finsler spaces
(a feature that is missing in Riemannian geometries because of quadratic, or
quartic constraints) is the property of *Torsion*. Chern has shown how to
interpret Finsler spaces in terms of the parametric independent calculus of
variations, where the dual field, V, is projectivized.

Cartan recognized in 1922 that spaces with torsion existed, but he achieved
this result from an entirely different tack than that used by Finsler. Cartan
hypothecated that in the sense that a structural defect of closure in a basis
frame implies the existence of curvature 2-forms, a structural defect of closure
in the position of the origin could imply the existence of torsion 2-forms. As
Brillouin (1959) states

**"If one does not admit the symmetry of the (connection) coefficients,
one obtains the twisted spaces of Cartan, spaces which scarcely have been used
in physics to the present, but which seem to be called to an important
role." . **

To this day not much has been done with Cartan's spaces with torsion, except in
the Kondo theory of dislocation defects in solids. However, Cartan's notions
can be made precise and more general in terms of subspaces of a projectivized
basis frame. Then not only the position of the origin (the point of perspective
in projective geometry), but also its dual polar axis can have closure defects,
leading to dislocations and disclinations respectively

(Kiehn 1994)

Details