Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 19; d.
Another relation of the Riemann tensor involves the covariant derivative of the tensor, and is known as the Bianchi identity (actually the second Bianchi identity; the first
identity is the symmetry
relation that we saw earlier). The identity is easiest to derive at the origin of a locally inertial frame (LIF), where the first derivatives of the metric tensor, and thus the Christoffel symbols, are all zero. At this point, we have
If the Christoffel symbols are all zero, then the covariant derivative becomes the ordinary derivative
Therefore, we get, at the origin of a LIF:
By cyclically permuting the index of the derivative with the last two indices of the tensor, we get
As usual we can use the argument that since we can set up a LIF with its origin at any non-singular point in spacetime, this equation is true everywhere and since the covariant derivative is a tensor, this is a tensor equation and is thus valid in all coordinate systems. This is the Bianchi identity.