How is the Riemann tensor proportinial to the curvature scalar?

[Lyalpha]

My professor asks, "Double check a formula that specifies how Riemann tensor is proportional to a curvature scalar." in our homework.

The closet thing I can find is the relation between the ricci tensor and the curvature scalar in einstein's field equation for empty space.

 

[Lyalpha]

And by proportinial, I mean proportional.

 

[haushofer]

He probably means the relation one has for maximally symmetric spaces, which you can find e.g. in Nakahara. It should be something like

<br /> R_{abcd} \propto R [g_{[a[c}g_{d]b]}]<br />

You can check this for a sphere, deSitter and antideSitter.