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Suppose that it is given that the Riemann curvature tensor in a special kind of spacetime of dimension $d\geq2$ can be written as $$R_{abcd}=k(x^a)(g_{ac}g_{bd}-g_{ad}g_{bc})$$ where $x^a$ is a vector in the space.

What condition can I impose on $d$ to guarantee that $k(x^a)$ is a constant? I have been told that such a condition exists.

Not knowing what to do, I computed the contractions of the curvature tensor and got $$R_{bd}=g^{ac}R_{abcd}=k(d-1)g_{bd}$$ and $$R=g^{bd}g^{ac}R_{abcd}=kd(d-1)$$ But it probably is of no use?

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I don't know if this works, but maybe you want to use the fact that the Einstein tensor is covariantly conserved. –  Prahar May 11 '13 at 13:13
    
@Prahar: Ahhhh!! thanks! you're a genius :P –  Clarice May 11 '13 at 16:39
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