Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question
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 $(\nabla f)_{eabcd}$.

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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.