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".