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For a riemannian metric $g$ consider the following tensor $T_{rstu}=k(x)g_{rt}g_{su}-k(x)g_{st}g_{ru}$. Which condition has to satisfy $k$ if we want the tensor $T$ to be the Riemann tensor of a Levi-Civita connection?

any suggestions? thanks

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If the Riemannian curvature tensor takes this form, then at any given point all the sectional curvatures are equal. Thus if $n>2$ then Schur's Lemma tells us that the sectional curvature is in fact constant; i.e. $k$ must be constant. If $n=2$ then this is the problem of prescribed Gauss curvature, which you should be able to find many papers about.

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