Linear Transformations on Affine-Connections

Understand

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.