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OP is considering Yang-Mills theory over a curved base space $(M,g)$. If the base space connection is the Levi-Civita connection $\nabla^{LC}=\partial+\Gamma$, then it doesn't matter whether one uses the gauge-covariant derivative $D=\partial+A$ or the full covariant derivative $\nabla=D+\Gamma$ as the Christoffel symbols $\Gamma$ drops out of the Yang-Mills ...



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