Types
Bianchi identity
The first identity
R i j k l + R i k l j + R i l j k = 0
The second identity
∇ l R m i j k + ∇ j R m i k l + ∇ k R m i l j = 0
Or ∇ j ( R j k − 1 2 g j k R ) = 0
Or R j k − 1 2 g j k R = k T j k , ∇ j T j k = 0