# What is the covariant derivative of a metric tensor $\nabla_{\mu} g^{\mu\nu}$ =?

What is the covariant derivative of a metric tensor, this particular one to be specific $$\nabla_{\mu} g^{\mu\nu}$$? Notice we've got repetitive indices here. Is it zero and has it got to do anything with $$\nabla_{\alpha} g^{\mu\nu}=0~?$$

Here $$\nabla_{\mu}$$ is the covariant derivative and the connection is given by the Christoffel symbol $$\Gamma^{\mu}_{\alpha \beta}$$.

Yes, it is zero. $$\nabla_{\alpha} g^{\mu\nu}$$ is a three-index tensor. If any tensor is zero, all of its contractions are zero.
To add a bit on the subject, when you don't assume that the connection needs to be metric-compatible, then you may have what is called non-metricity, defined as follows: $$Q_{\alpha \beta \gamma}\equiv \nabla_{\alpha}g_{\beta \gamma}$$.
And then the object you are mentioning is no longer zero but rather a non-vanishing vector $$\hat{Q}^\alpha$$, where $$\hat{Q}^\alpha \equiv g_{\mu \nu}Q^{\mu \nu \alpha} \propto \nabla_\mu g^{\mu \nu}$$