The Ricci tensor

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Before looking at the consequences of the Bianchi identities, we need to define the Ricci tensor  :

It is the contraction of on the first and third indices. Other contractions would in principle also be possible: on the first and second, the first and fourth, etc. But because is antisymmetric on and and on and , all these contractions either vanish or reduce to . Therefore the Ricci tensor is essentially the only contraction of the Riemann tensor.

Similarly, the Ricci scalar  is defined as