Lovelock gravity

1. Sep 12, 2011

alejandrito29

http://en.wikipedia.org/wiki/Lovelock_theory_of_gravity

I dont undestand how generate a Riemman tensor $$R_{ijkl}$$ in the expression:

$$\delta^{abcd}_{ABCD}R^{AB}_{ab}R^{CD}_{cd} = R_{ijkl}R^{ijkl}-4R_{ij}R^{ij}+R^2$$
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$$\delta^{abcd}_{ABCD}$$ is the generalized kronecker delta,

i understan that a term

$$\delta^a_A \delta^b_B \delta^c_C \delta^d_D R^{AB}_{ab}R^{CD}_{cd} = R^2$$

but , for example
$$\delta^a_A \delta^b_B \delta^c_D \delta^d_C R^{AB}_{ab}R^{CD}_{cd} ??$$

for example , how i find the term $$R_{ijkl}$$ ?

Last edited: Sep 12, 2011
2. Feb 14, 2012