Tensor notation of covariant derivative

Question

I'm trying to apply Wald's General Relativity equation $3.1.14$:
$$nabla_a{T^{b_1dots b_k}}_{c_1dots c_{ell}}=overline{nabla}_a{T^{b_1dots b_k}}_{c_1dots c_{ell}}+sum_i{C^{b_i}}_{ad}{T^{b_1dots ddots b_k}}_{c_1dots c_{ell}}-sum_j{C^d}_{ac_j}{T^{b_1dots b_{ell}}}_{c_1dots ddots c_{ell}}$$
to the covariant derivative of a 2-form, and I'm having trouble understanding what the notation "$dots ddots$" means.
I am familiar with Einstein's summation convention. What is unclear to me is what is the position of the "$d$" index. Does any position work?

Eletie · Answer

$d$ is just the summed (dummy) index. In words, the procedure would be as follows: in the first summation, replace an upper index in $T$ with $d$, and use the replaced index in the object $C$, along with the lower dummy index $d$ to sum over. The final lower index of $C$ is always the index from the covariant derviative $nabla$. You repeat this until you've covered all the upper indices in $T$. E.g. the simplest case for a vector is just
$$ nabla_a T^b = bar{nabla}_a T^b + C^b{}_{ad} T^d  . $$
And then you can work out the analogous procedure for the lower indices of $T$.

Tensor notation of covariant derivative

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