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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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4  
Yes, you have to sum over $c$ as well. Why are you hesitating? –  Raskolnikov Feb 21 '12 at 11:57
1  
Why do you feel $c$ is different from $a,b$? –  anon Feb 21 '12 at 12:16
    
@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! –  harlekin Feb 23 '12 at 23:46
    
@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! –  harlekin Feb 23 '12 at 23:48
    
Oh, I see. Even when double-indices occur in a single symbol/tensor, it still signifies implicit summation. :) –  anon Feb 23 '12 at 23:50

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