curvature tensor and the metric

Question

how does

$ R_{βδ} = g^{αγ}R_{αβγδ} $

but

$g^{αβ}R_{αβγδ}=0$

I am unsure of the methods used although I have seen the symmetries of the curvature tensor

Answer 1

The indices being contracted are not the same, the curvature tensor is antisymmetric under the exchange of $\alpha$ and $\beta$, but not under the exchange of $\alpha$ and $\gamma$. The argument that the second contraction gives zero is just that a symmetrical matrix (as the one representing $g$) times an asymmetrical one is $0$, and this obviously only works if the indices of the second matrix actually are asymmetric under exchange.