Proving Constant Curvature in n-Dimensional Manifold

[camipol89]

Hello everybody,
How do you prove that,given an n-dimensional manifold with constant curvature , i.e.

the constant K is given by : K= R/n(n-1) (R denotes the scalar curvature)?

I tried to contract the Riemann tensor in the expression above to obtain on the left side the scalar curvature but the right side vanishes :(
What am I doing wrong?
Thanks

 

[camipol89]

Sorry if you have to click on the image to actually see it but I don't know how to write the riemann tensor in Latex code...