In this work, we show that a class of metric-affine gravities can be reduced to a Riemann-Cartan one. The reduction is based on the cancelation of the nonmetricity against the symmetric components of the spin connection. A heuristic proof, in the Einstein-Cartan formalism, is performed in the special case of diagonal unitary tangent metric tensor. The result is that the nonmetric degrees of freedom decouple from the geometry. Thus, from the point of view of isometries on the tangent manifold, the equivalence might be viewed as an isometry transition from the affine group to the Lorentz group, A(d,R) → SO(d). Furthermore, in this transition, depending on the form of the starting action, the nonmetricity degrees might present a dynamical matter field character, with no geometric interpretation in the Riemann-Cartan geometry.
Differential geometry; Nonmetricity; Gauge theories of gravity; Metric-affine gravity
