The Basic Simplicity of ECE Theory

The basic structure of ECE theory is the basic structure of Cartan’s differential geometry. This structure consists of the two Maurer Cartan structure equations:

T = d ^ q + omega ^ q


R = d ^ omega + omega ^ omega

in shorthand notation explained on this website. Here T is the torsion form, q is the tetrad form, omega is the spin connection, R is the curvature form. Given these structure equations, Cartan derived a well known identity:

d ^ T + omega ^ T := R ^ q

and this is proven in several papers of this website (notably paper 102, eq. (9.20)). Eq. (9.20) breaks out the Cartan identity into tensor notation, showing that one side of the identity is precisely identical to the other side, a self checking proof. This proof is usually an exercise for students in differential geometry. It is written out in paper 102 for the non-specialist. There also exists the Evans identity, which is

d ^ T* + omega ^ T* := R* ^ q

where * is the well known Hodge dual. The Evans identity is proven in full detail in paper 137, Eq. (30), which precisely parallels eq. (9.20) of paper 102. The Evans identity is a precise identity of differential geometry.

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