Although the autoparallel curves and the geodesics coincide in the Riemannian geometry in which only the curvature is nonzero among the nonmetricity, the torsion and the curvature, they define different curves in the non-Riemannian ones. We give a novel approach to autoparallel curves and geodesics for theories of the symmetric teleparallel gravity written in the coincident gauge. Then we apply our autoparallel equation to a Schwarzschild-type metric and give remarks about dark matter and orbit equation.

After summarizing the basic concepts for the exterior algebra, we first discuss the gauge structure of the bundle over base manifold for deciding the form of the gravitational sector of the total Lagrangian in any dimensions. Then we couple minimally a Dirac spinor field to our gravitational Lagrangian 2-form which is quadratic in the nonmetricity and both linear and quadratic in the curvature in two dimensions. Subsequently, we obtain field equations by varying the total Lagrangian with respect to the independent variables. Finally, we find some classes of solutions of the vacuum theory and then a solution of the Dirac equation in a specific background and analyze them.

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.