Understanding Lovelock Gravity Theory and Riemman Tensor

[alejandrito29]

I read about lovelock gravity:
http://en.wikipedia.org/wiki/Lovelock_theory_of_gravity

I don't undestand how generate a Riemman tensor $$R_{ijkl}$$ in the expression:

$$\delta^{abcd}_{ABCD}R^{AB}_{ab}R^{CD}_{cd} = R_{ijkl}R^{ijkl}-4R_{ij}R^{ij}+R^2$$
`
$$\delta^{abcd}_{ABCD}$$ 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 $$R_{ijkl}$$ ?

 

[ahmadfouad]

use the defination of the generalized kronecker delta as determinant

 

[Bill_K]

In your second example all you did was switch C and D. The Riemann tensor is antisymmetric on C and D so this term is just -R². To get a term like R^abcdR_abcd you would need to use for example R^CD_abR^AB_cd.