  In the conventional formulation of general relativity, gravity is represented
by the metric curvature of Riemannian geometry. There are also alternative
formulations in flat affine geometries, wherein the gravitational dynamics is
instead described by torsion and nonmetricity. These so called general
teleparallel geometries may also have applications in material physics, such as
the study of crystal defects. In this work, we explore the general teleparallel
geometry in the language of differential forms. We discuss the special cases of
metric and symmetric teleparallelisms, clarify the relations between
formulations with different gauge fixings and without gauge fixing, and develop
a method of recasting Riemannian into teleparallel geometries. As illustrations
of the method, exact solutions are presented for the generic quadratic theory
in 2, 3 and 4 dimensions.
