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