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I'm reading Zee's Einstein Gravity in a Nutshell and while calculating the Ricci tensor for FRW spacetime he claims that because the spacelike slices of constant $t$ are rotationally invariant, the off-diagonal elements of the Ricci tensor vanish (working in $t$, $r$, $\theta$, $\phi$ coordinates). Without calculating the Ricci tensor from the Christoffel symbols element-by-element, can someone provide an explanation of why this is true? More generally is there a good way to see how symmetries of the metric, like rotation or time translation, etc. translate to properties of the Ricci or Einstein tensor?

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