# What is the Riemann curvature tensor contracted with the metric tensor?

Can the Ricci curvature tensor be obtained by a 'double contraction' of the Riemann curvature tensor? For example

$R_{\mu\nu}=g^{\sigma\rho}R_{\sigma\mu\rho\nu}$.

-

I'm not sure what you mean by 'double contraction', but the Ricci tensor in local coordinates is given by \begin{align} R_{\mu \nu} = R^\rho_{~~\mu \rho \nu}, \end{align} which is the same as $g^{\sigma \rho} R_{\sigma \mu \rho \nu}$, exactly what you have written.

-
It's been a long day... cheers buddy. –  user12345 Feb 1 '13 at 21:22
Yes. The expression for the Ricci tensor is often written as (see here) $$R_{\mu\nu} = R^{\alpha}_{\phantom\alpha \mu\alpha\nu},$$ but the right hand side is precisely what you wrote since the metric simply raises the first index.
You still get +1 from me anyway. Also, since we know that $R_{\mu\nu}=R_{\nu\mu}$, would it be true to say that we can swap the first and third indices in the Riemann tensor, even if they're all downstairs? –  user12345 Feb 1 '13 at 21:45