Can the Ricci curvature tensor be obtained by a 'double contraction' of the Riemann curvature tensor? For example
$R_{\mu\nu}=g^{\sigma\rho}R_{\sigma\mu\rho\nu}$.
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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. |
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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. |
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