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I am trying to understand the expression for Scalar curvature in terms of the Christoffel symbols.

This is given on Wikipedia by \begin{equation} S = g^{ab}(\Gamma^c_{ab,c} - \Gamma^c_{ac,b} + \Gamma^d_{ab}\Gamma^c_{cd} - \Gamma^d_{ac}\Gamma^c_{bd}) \end{equation} (see here for the Wikipedia post)

The thing I'm not sure about is the comma in expression $ab,c$, for example in the footer of the first Symbol. My guess would be that it means \begin{equation} \Gamma^c_{ab,c} = \partial_c \Gamma^c_{ab} \end{equation}

Is that correct ? The site on Christoffel Symbols on Wikipedia doesn't explain what the comma means so I was wondering whether somebody could help? Many thanks!

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up vote 6 down vote accepted

In differential geometry, a subscript with a comma often is used to denote the coordinate partial derivative relative to some fixed coordinate system.

Along the same lines, a subscript with a semi-colon is often used to denote the covariant derivative.

So, given a scalar function $f:M\to\mathbf{R}$,

$$ \nabla_a f = \partial_a f = f_{,a} = f_{;a} $$

For a tensor quantity with coordinate components $f_{abcd}$, the expression $ f_{abcd,e}$ means the $\partial_e$ of the scalar function $f_{abcd}$. Which is different from $f_{abcd;e}$ which often means the scalar component $(\nabla f)_{eabcd}$.

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that's great help, thanks also for giving this a little context! –  harlekin Feb 2 '12 at 12:16

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