Proving Constant Curvature in n-Dimensional Manifold

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The discussion focuses on proving that in an n-dimensional manifold with constant curvature, the constant K is defined by the equation K = R/n(n-1), where R represents the scalar curvature. A participant expresses difficulty in contracting the Riemann tensor to derive the scalar curvature, noting that their attempts result in a vanishing right side. This indicates a misunderstanding or misapplication of the Riemann tensor properties in relation to scalar curvature calculations.

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camipol89
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Hello everybody,
How do you prove that,given an n-dimensional manifold with constant curvature , i.e.

2722313-0.png


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
 
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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...
 

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