  After recalling the differential geometry of non-metric connections in the
formalism of differential forms, we introduce the idea of a Non-Metricity (NM)
connection, whose connection $1$--forms coincides with the non-metricity
$1$--forms for a class of cobase fields. Then we formulate a theory of
gravitation (equivalent to General Relativity (GR)) which admits a geometrical
interpretation in a flat torsionless space where the gravitational field is
completely manifest in the non-metricity of a NM connection. We define and then
apply the non-metricity gauge to a gravitational Lagrangian density discovered
by Wallner and which is equivalent to the Einstein-Hilbert Lagrangian density.
The Einstein equations coupled to the matter currents $\left(
\mathcal{J}_{\alpha}\right) $ thus becomes $\delta
dg_{\alpha}=\mathcal{T}_{\alpha}+\mathcal{J}_{\alpha}$, where $\left(
\mathcal{T}_{\alpha}\right) $ is identified as the gravitational
energy-momentum currents, to which we shall find a relatively simple and
physically appealing form. It is also shown that in the gravitational analogue
of the Lorenz gauge, our field equations can be written as a system of Proca
equations, which may be of interest in the study of propagation of
gravitational-electromagnetic waves.
