I am not sure how to interpret the following expression with regard to the Einstein summation convention
\begin{equation} g^{ab}(\partial_c \Gamma^c_{ab} - \partial_b \Gamma^c_{ac}) \end{equation}
(It's not important for the question, but $g$ here is the metric on a Riemannian manifold, $\Gamma$ are the Chritstoffel symbols and $\partial_c = \frac{\partial}{\partial x_c}$.)
Do I have to sum here over $c$ as well ?
So if I write the above out using the summation sign, is the following correct? \begin{equation} g^{ab}(\partial_c \Gamma^c_{ab} - \partial_b \Gamma^c_{ac}) = \sum_{a,b} \left(\sum_c g^{ab}(\partial_c \Gamma^c_{ab} - \partial_b \Gamma^c_{ac})\right) \qquad \end{equation}
Thanks for your help!