How many independent equations are contained in $R_{rsmn}=0$ in consideration of the Bianchi identity? In $d$ dimensions, how many independent equations are contained in $R_{rsmn}=0$ in consideration of the Bianchi identity $\nabla_{[a}R_{bc]de}=0$?
This discussion reveals the independent equations contained in the Bianchi identity in consideration of the symmetries of the Riemann tensor, but I'm not quite sure how to use that number for this question.
I've given this a great deal of thought, but somehow, I don't seem to be making progress.
 A: First, note that in a $d$ dimensional spacetime, the Riemann tensor has $\frac{d^2(d^2-1)}{12}$ independent components, using the symmetries $R_{abcd}=-R_{bacd}=-R_{abdc}=R_{cdab}$ and Bianchi identity $R_{a[bcd]}=0$. (see eg Counting independent components of the Riemann curvature tensor).
Then I claim $\frac{d^2(d^2-1)}{12}$ is also the answer to your question. Here is my argument.

*

*If you want to set $R_{abcd}=0$ at a point $x$, then you need to set $\frac{d^2(d^2-1)}{12}$ components to zero. The derivative of the Riemann tensor $\nabla_a R_{bcde}$ at the point $x$ is an independent rank-5 tensor.

*If you want to set $R_{abcd}=0$ everywhere in the spacetime, then again using the freedom to specify that $\frac{d^2(d-1)^2}{12}$ components are zero everywhere is sufficient. The reason is that once the Riemann tensor vanishes everywhere, then you are dealing with a flat space, and the gradient of the Riemann tensor is automatically zero, so the Bianchi identify $\nabla_{[a} R_{bc]de}=0$ is automatically satisfied.

