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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!

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    Yes, you have to sum over $c$ as well. Why are you hesitating?2012-02-21
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    Why do you feel $c$ is different from $a,b$?2012-02-21
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    @anon Sorry for the late reply, I was unsure because c in the second term is only in the Christoffel symbol - looking back at it with your feedback I think my question was stupid, really! Thks for your comment!2012-02-23
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    @Raskolnikov sorry for the late answer, I was hesitating because the c in the second term is not a pairing of symbols (so to speak), instead I have to sum within the Christoffel expression. But now I think I would not hesitate to do that, thanks to your comment!2012-02-23
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    Oh, I see. Even when double-indices occur in a single symbol/tensor, it still signifies implicit summation. :)2012-02-23

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