Does the trace of a Levi-Civita connection always vanish?

Question

If $\Gamma^a_{bc}$ are the Christoffel symbols of the Levi-Civita connection corresponding to the metric tensor $\boldsymbol{g}$, i.e., if $$\Gamma^a_{bc} = \tfrac{1}{2} \, g^{ad} (g_{cd,b} + g_{bd,c} - g_{bc,d}),$$ does it follow that $\Gamma^b_{bc} = 0$? Thank you for any help you can give.