# Einstein-Yang-Mills Connections

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 $\nabla=D+\Gamma$ since the Christoffel symbols $\Gamma$ drops out of the Yang-Mills theory and OP's eqs. (3), (4) and (5). This is mainly due to the torsionfreeness of the Levi-Civita connection $\nabla^{LC}$.