Linear transformations on affine-connections

We state and prove a simple theorem that allows one to generate invariant quantities in metric-affine geometry, under a given transformation of the affine connection. We start by a general functional of the metric and the connection and consider transformations of the affine connection possessing a certain symmetry. We show that the initial functional is invariant under the aforementioned group of transformations iff its Gamma-variation produces tensor of a given symmetry. Conversely if the tensor produced by the Gamma-variation of the functional respects a certain symmetry then the functional is invariant under the associated transformation of the affine connection. We then apply our results in metric-affine gravity and produce invariant actions under certain transformations of the affine connection.