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
