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 $Γ$-variation produces tensor of a given symmetry. Conversely if the tensor produced by the $Γ$-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. Finally, we derive the constraints put on the hypermomentum for such invariant Theories.

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