Tensor notation of covariant derivative

Question

I'm trying to apply Wald's General Relativity equation $3.1.14$:

$$\nabla_a{T^{b_1\dots b_k}}_{c_1\dots c_{\ell}}=\overline{\nabla}_a{T^{b_1\dots b_k}}_{c_1\dots c_{\ell}}+\sum_i{C^{b_i}}_{ad}{T^{b_1\dots d\dots b_k}}_{c_1\dots c_{\ell}}-\sum_j{C^d}_{ac_j}{T^{b_1\dots b_{\ell}}}_{c_1\dots d\dots c_{\ell}}$$

to the covariant derivative of a 2-form, and I'm having trouble understanding what the notation "$\dots d\dots$" 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?

Answer 1

$d$ 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 $T$.