Computing Ricci scalar from 3D metric with diagonal components

[binbagsss]

Homework Statement

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I am trying to compute $R$ from the 3-d metric: $ds^{2}=d\chi^{2}+f^{2}\chi(d\theta^{2}+sin^{2}\theta d\phi^{2})$

Homework Equations

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The space also satisfies the below relationships:
$R=3k$
$R_{abcd}=\frac{1}{6}R(g_{ac}g_{db}-g_{ad}g_{bc})$ [1]

The Attempt at a Solution

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I think I need to compute the christoffel symbols, then the Riemann tensor, and contract etc.
I'm just wondering whether the task is meant to be simplified by eq [1]?( I can see the metric is diagonal and so this reduces the number of non-zero christoffel symobols...)

Thanks in advance.

 

[Matterwave]

What is "k" in equation 1?