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I am trying to do the variation of Gauss Bonnet Invariant, and the Gauss Bonnet Invariant is:

$G$=$R^2$+$R_{abcd}$$R^{abcd}$-$4R^{ab}$$R_{ab}$

The variation of $G$ is:

$\delta$$G$=$2R\delta$$R$+ $\delta($$R_{abcd}$$R^{abcd}$)-$\delta$$(4R^{ab}$$R_{ab}$)

I am having problem in doing the variation of $\delta($$R_{abcd}$$R^{abcd}$).

Can anyone please give me the solution in detail? I have the answer but I don't know how to solve it.

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You will need \begin{equation} \begin{split} \delta \Gamma^c_{ab} &= \frac{1}{2} \left( \nabla_a h_b{}^c + \nabla_b h_a{}^c - \nabla^c h_{ab} \right) ~, \\ \delta R^a{}_{bcd} &= \frac{1}{2} \nabla_c \nabla_d h_b{}^a + \frac{1}{2} \nabla_c \nabla_b h_d{}^a - \frac{1}{2} \nabla_c \nabla^a h_{db} -\frac{1}{2} \nabla_d \nabla_c h_b{}^a -\frac{1}{2} \nabla_d \nabla_b h_c{}^a + \frac{1}{2} \nabla_d \nabla^a h_{cb} ~, \\ \delta R_{ab} &= \frac{1}{2} \left( \nabla_c \nabla_a h_b{}^c + \nabla_c \nabla_b h_a{}^c - \nabla^2 h_{ab} - \nabla_a \nabla_b h \right) ~, \\ \delta R &= - R_{ab} h^{ab} + \nabla_a \nabla_b h^{ab} - \nabla^2 h ~, \\ \delta \det g &= h \det g ~. \\ \end{split} \end{equation} where $h_{ab} = \delta g_{ab}$ and $h=g^{ab} h_{ab}$.

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    $\begingroup$ I actually know this but what next? $\endgroup$ – user2129579 Nov 15 '17 at 14:28
  • $\begingroup$ My question is that how to solve $\delta($$R_{abcd}$$R^{abcd}$) further? In solution I have something like $-4R_{acbd}$$R^{cd}$ .... etc $\endgroup$ – user2129579 Nov 15 '17 at 14:35
  • $\begingroup$ Can you please give me the detail of it? I am actually trying to derive the field equations of $f(R,G)$ gravity and I am stuck at this point $\endgroup$ – user2129579 Nov 15 '17 at 14:36
  • $\begingroup$ +1, the curvature at any local point is indeed the global invariance of divergence. It also satisfies Ergodium Thereom with the determinants at the end. $\endgroup$ – user97261 Nov 15 '17 at 17:05

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