# Action of connection on connection of metric tensor

Let $$\nabla$$ be levi civita connection on Riemannian manifold $$M$$. I was wondering, what is $$\nabla_{\alpha}(\partial_{\beta}g_{\mu\nu})$$?

Is it $$\partial_{\alpha}\partial_{\beta}g_{\mu \nu}-\Gamma^{\sigma}_{\alpha \beta}\partial_{\sigma}g_{\mu\nu}-\Gamma^{\sigma}_{\alpha \nu}\partial_{\beta}g_{\sigma \nu}-\Gamma^{\sigma}_{\alpha \nu}\partial_{\beta}g_{\mu \sigma}$$?

This is a weird expression to me, because $$\partial_{a}g_{bc}$$ is not a tensor, and so applying a covariant derivative to it is not natural. Generally, one would evaluate double covariant derivatives by going from "outside" in, so that you are are always evaluating tensors. So, for example, for some tensor $$T_{ab}$$, you have:
$$\nabla_{c}\nabla_{d}T_{ab} = \partial_{c}\left(\nabla_{d}T_{ab}\right) - \Gamma_{cd}{}^{e}\nabla_{e}T_{ab} - \Gamma_{ca}{}^{e}\nabla_{d}T_{eb} - \Gamma_{cb}{}^{e}\nabla_{e}T_{ae}$$
And you then expand out the rest with the ordinary expression for $$\nabla_{a}T_{bc}$$ and applying the ordinary product rule for partial derivatives
• what does $T^{\sigma}_{\beta}g_{\sigma \mu}$ evaluate to? Commented Dec 9, 2021 at 22:00
• You've lost the order of the indices the way you wrote that, but generally, the raising and lowering of indices on some tensor T is defined so that $$T_{ab} = g_{ac}T^{c}{}_{b} = T_{a}{}^{c}g_{cb}$$. Of course, if T is symmetric, the order doesn't matter. Commented Dec 9, 2021 at 22:39