I have worked out the quantum Bianchi idenities for the last two Covariant indices.

So... what is the Bianchi identity?

The first term doesn't matter, the indice can be covariant or contravariant. The identity satisfies sign changes in the order of the last three indices

[math]R_{\sigma \rho i j} + R_{\sigma ij \rho} + R_{\sigma j \rho i} = 0[/math]

To create the first Bianchi term is easy, contract a definition of the curvature with the metric tensor

[math]R_{\sigma \rho ij} = R^{k}_{\sigma ij} g_{\rho k}[/math]

Using the same principles, you can form the other three, now all three are,

[math]R_{\sigma \rho ij} = R^k_{\sigma ij} g_{\rho k}[/math]

[math]R_{\sigma i j \rho} = R^k_{\sigma j \rho} g_{i k}[/math]

[math]R_{\sigma j \rho i} = R^k_{\sigma \rho i} g_{j k}[/math]

There is a commutation relationship it seems between the metric and the connections. Using the formulation above though, we can see an equivalent form then of the Bianchi identity is

[math]R^k_{\sigma ij} g_{\rho k} + R^k_{\sigma j \rho} g_{i k} + R^k_{\sigma \rho i} g_{j k} = 0[/math]

Or simply as

[math]R_{\sigma [\rho i j]} = 0[/math]

In which the bracket denotes the antisymmetric part of the tensor, which arises from the antisymmetry in [math]i[/math] and [math]j[/math] and the definition of [math]\rho[ /math] is entangled into it.

It's also nice to note, this last identity can be seen to be related to the Jacobi triple vector product

[math]a \times (b \times c) + b \times (c \times a) + c \times (a \times b ) = 0[/math]

In which case, we understand it from the following relationship using basis vectors:

[math]\partial[\sigma,[\partial_\rho, \partial_i]] = 0[/math]

Now we can move on to the three important identities we looked at and they will give a quantized look at the identities.

[math]R_{\sigma \rho [i j]}g^{\sigma \rho} = \partial_i \Gamma_{j} - \partial_j \Gamma_{i} + \Gamma_{i} \Gamma_{j} - \Gamma_{j} \Gamma_{i}[/math]

[math]R_{\sigma i[j \rho]}g^{\sigma i} = \partial_j \Gamma_{\rho} - \partial_{\rho} \Gamma_{j} + \Gamma_{j} \Gamma_{\rho} - \Gamma_{\rho} \Gamma_{j}[/math]

[math]R_{\sigma j [\rho i]}g^{\sigma j} = \partial_{\rho} \Gamma_{i} - \partial_i \Gamma_{\rho} + \Gamma_{\rho} \Gamma_{i} - \Gamma_{i} \Gamma_{\rho}[/math]

You can write these three relationships out in the Bianchi identity, we can write the commutation again, on the indices

[math]R_{\sigma \rho[ i j]} + R_{\sigma i [j \rho]} + R_{\sigma j [\rho i]} = 0[/math]

The order of the commutation with respect to indices we have looked at have revealed their commutation in the last two indices.

math

**Edited by Dubbelosix, 08 September 2017 - 01:44 PM.**