# Expanding a summation of covariant derivatives

I hope this is not a silly question but I am trying to understand how this part of the equation works:

$$\nabla_{\lambda} \left( \nabla_{\mu}(R_{\nu \lambda}) + \nabla_{\nu}(R_{\mu \lambda}) \right)$$

Where $R_{xx}$ is the Ricci tensor.

My question: Am I required to work out the covariant derivatives separate then work out the 'outer' covariant derivative of the result or should I use the double covariant derivative rule twice like so:

\begin{align} \nabla_a \nabla_b h_{cd} &= \partial_a ( \partial_b h_{cd} - \Gamma^f_{bc} h_{fd} - \Gamma^f_{bd} h_{cf} ) \\ &\qquad \qquad - \Gamma^e_{ab} ( \partial_e h_{cd} - \Gamma^f_{ec} h_{fd} - \Gamma^f_{ed} h_{cf} ) \\ &\qquad \qquad - \Gamma^e_{ac} ( \partial_b h_{ed} - \Gamma^f_{be} h_{fd} - \Gamma^f_{bd} h_{ef} ) \\ &\qquad \qquad - \Gamma^e_{ad} ( \partial_b h_{ce} - \Gamma^f_{bc} h_{fe} - \Gamma^f_{be} h_{cf} ) \end{align}

• I think the result is a tensor, which is a vector space homomorphism I believe, so both ways should be the same? – Emil Feb 7 '18 at 20:06
• My Mathematica code is generating different results that's why. I am trying to derive the components of the field equations for $f(G)$ gravity – Mark Pace Feb 7 '18 at 20:25
• If they are not a homomorphism, I would say the inner covariant derivatives first, then the plus, then the outer covariant derivative, then the trace over lambda. Oh wait a moment, the lambdas are doubly indexed covariant? Is that a bug? – Emil Feb 8 '18 at 6:46
• Lamda is a free index in this instance – Mark Pace Feb 8 '18 at 7:49
• what Mathematica package are you using, if any ? – magma Feb 8 '18 at 20:18

## 1 Answer

The answer is that yes it is homomorphism and one can simply expand the brackets accordingly.

\begin{align} \nabla_a \nabla_b h_{cd} &= \partial_a ( \partial_b h_{cd} - \Gamma^f_{bc} h_{fd} - \Gamma^f_{bd} h_{cf} ) \\ &\qquad \qquad - \Gamma^e_{ab} ( \partial_e h_{cd} - \Gamma^f_{ec} h_{fd} - \Gamma^f_{ed} h_{cf} ) \\ &\qquad \qquad - \Gamma^e_{ac} ( \partial_b h_{ed} - \Gamma^f_{be} h_{fd} - \Gamma^f_{bd} h_{ef} ) \\ &\qquad \qquad - \Gamma^e_{ad} ( \partial_b h_{ce} - \Gamma^f_{bc} h_{fe} - \Gamma^f_{be} h_{cf} ) \end{align}

May be used in this instance