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Historical Background

After the dicovery of GR in 1915/16, H. Weyl was the first trying to extend riemannian geometry in order to describe electromagnetism and gravity in a unified language. He used nonmetric connections and the tensor of nonmetricity, but his attempt failed. In 1922, the famous french mathematician E. Cartan came out with a extension of riemannian geometry using a connections that were not neccessarily symmetric in the lower two indices. He suspected that the tensor obained in this way, called Cartan's torsion now, could be relevant for electrodynamics. Very interesting in this context is the book Einstein-Cartan, Letters on absolute parallelism 1929-1932 by Debever. Hereafter, Einstein published a series of articles in the session reports of the Prussian Academy of Sciences about distant parallelism and a unified field theory. The above article recapitulates these papers proposing field equations that include the torsion tensor and yield in first approximation both Maxwell's equations and Newton-Poisson equations. The name distant parallelism cames from the fact that in this theory the Riemannian curvature tensor (to be distinguished from the Riemann-Christoffel curvature tensor used in GR) vanishes everywhere.

Unfortunately, Einstein did never never try to incorporate quantum mechanics into this picture, nor the other physicits, as he complained, did support his efforts extending the GR formalism. So this approach -lacking a description of particles- had to fail.

Nevertheless, the theory is beautiful and was essential for the development of modern differential geometry. An excellent book that treats the tensor calculus is

J.A.Schouten, Ricci-Calculus (Springer, 2nd ed. 1954), in particular chap. 3 "Linear connexions".