I am playing around with coupling a classical $SU(2)$ Yang-Mills theory to Einstein's equations.
Assuming spherical symmetry, the $SU(2)$ connection can be written \begin{equation} A = \omega(r)\tau_1 d\theta + \omega(r)\sin\theta \tau_2 d\theta + \cos\theta \tau_3 d\phi,\tag{1} \end{equation} where the $\tau_i$ are the generators of the $\mathfrak{su}(2)$ algebra.
A static spherically symmetric metric has the form \begin{equation} ds^2 = -T^{-2}(r)dt^2 + B^{-1}(r)dr^2 + r^2d\theta^2 + r^2\sin^2\theta d\phi^2.\tag{2} \end{equation}
The Yang-Mills equations are \begin{equation} D\star F = 0,\tag{3} \end{equation} along with the Bianchi Identity \begin{equation} DF = 0.\tag{4} \end{equation}
Clearly, $D$ contains the usual gauge covariant exterior derivative \begin{equation} DF = dF + [A\wedge F]\tag{5} \end{equation} with respect to the $SU(2)$ connection.
Here is where my question arises: since the Yang-Mills field lives in curved spacetime, shouldn't the gauge covariant exterior derivative include additional terms which describe the usual covariant derivative of $F$ with respect to the Levi-Civita connection on the spacetime manifold?