I'm trying to prove that under the Ricci flow, the Ricci tensor evolves by the following equation:
$$\frac{\partial}{\partial t} R_{i k}=\Delta R_{i k}+2 g^{p q} g^{r s} R_{p i k r} R_{q s}-2 g^{p q} R_{i p} R_{q k}$$
This is corollary $4.18$ of Ben Andrew's "The Ricci Flow in Riemannian Geometry" book. Now, the proof they give there starts like this:
Where $(4.9)$ is given by:
$$\frac{\partial}{\partial t} g^{i j}=-g^{i k} g^{j \ell} h_{k \ell}$$
and Theorem $4.14$ is:
$$\begin{aligned} \frac{\partial}{\partial t} R_{i j k \ell}=& \Delta R_{i j k \ell}+2\left(B_{i j k \ell}-B_{i j \ell k}-B_{i \ell j k}+B_{i k j \ell}\right) \\ &-g^{p q}\left(R_{p j k \ell} R_{q i}+R_{i p k \ell} R_{q j}+R_{i j k p} R_{q \ell}+R_{i j p \ell} R_{q k}\right) \end{aligned}$$
where
$$B_{i j k \ell}=g^{p r} g^{q s} R_{p i q j} R_{r k s \ell}=R_{i j}^{p q} R_{p k q \ell}$$
Now, I see where all the terms in my first picture (with the red outlined term) are coming from. I understand they ought to be here. But the term $-2g^{jp}g^{lq}R_{pq}R_{ijkl}$ outlined in red just vanished! And the authors never mention it again. What happened here? I've been staring at this for a long while now but can't figure this out. I would be really grateful for any help. I think this is probably a mistake on my part, because in Hamilton's $1982$ original paper, the same "mistake" is there, so I think there must be something going on here that I'm not seeing.
Thanks in advance!
UPDATE: I think a mistake was indeed committed by the authors, but overall the work is correct. First of all, the term I outlined in red has the wrong sign, it should be ${+2g^{jp}g^{lq}R_{pq}R_{ijkl}}$ rather than $\color{red}{-2g^{jp}g^{lq}R_{pq}R_{ijkl}}$, as one can easily check (and in Hamilton's original paper the sign is correct). Secondly, the authors later compute that:
$$\begin{array}{l} 2 g^{j \ell}\left(B_{i j k \ell}-B_{i j \ell k}-B_{i \ell j k}+B_{i k j \ell}\right) \\ \quad=2 g^{j \ell} B_{i j k \ell}-2 g^{j \ell}\left(B_{i \ell j k}+B_{i j \ell k}\right)+2 g^{p r} g^{q s} R_{p i q k} R_{r s} \\ \quad=2 g^{j \ell} B_{i j k \ell}-4 g^{j \ell} B_{i j \ell k}+2 g^{p r} g^{q s} R_{p i q k} R_{r s} \\ \quad=2 g^{j \ell}\left(B_{i j k \ell}-2 B_{i j \ell k}\right)+2 g^{p r} g^{q s} R_{p i q k} R_{r s} \end{array}$$
and
$$\begin{array}{l} g^{j \ell} g^{p q}\left(R_{p j k \ell} R_{q i}+R_{i p k \ell} R_{q j}+R_{i j p \ell} R_{q k}+R_{i j k p} R_{q \ell}\right) \\ \quad=2 g^{p q} R_{p i} R_{q k}+g^{j \ell} g^{p q} R_{i p k \ell} R_{q j}+g^{j \ell} g^{p q} R_{i j k p} R_{q \ell} \\ \quad=2 g^{p q} R_{p i} R_{q k}+2 g^{p r} g^{q s} R_{p i q k} R_{r s} \end{array}$$
Therefore, now taking into account the term I said was missing, we have:
$$\begin{aligned} \frac{\partial}{\partial t} R_{i k}&=\Delta R_{i k}+2 g^{j \ell}\left(B_{i j k \ell}-2 B_{i j \ell k}\right)+2 g^{p r} g^{q s} R_{p i q k} R_{r s}-2 g^{p q} R_{p i} R_{q k} - 2 g^{p r} g^{q s} R_{p i q k} R_{r s} \color{red}{ +2g^{jp}g^{lq}R_{pq}R_{ijkl}}\\ &= \Delta R_{i k}+2 g^{j \ell}\left(B_{i j k \ell}-2 B_{i j \ell k}\right) \color{red}{ +2g^{jp}g^{lq}R_{pq}R_{ijkl}} -2 g^{p q} R_{p i} R_{q k} \end{aligned}$$
So all we have to do now is prove that:
$$2g^{jp}g^{lq}R_{pq}R_{ijkl} = 2 g^{p r} g^{q s} R_{p i q k} R_{r s}$$
Which is indeed true, since:
$$g^{p r} g^{q s} R_{p i q k} R_{r s} = R_{rs} R((e^r)^{\#}, e_i, (e^s)^{\#}, e_k) = R_{pq} R((e^p)^{\#}, e_i, (e^q)^{\#}, e_k)$$
and
$$g^{jp}g^{lq} R_{pq} R_{ijkl} = R_{pq} R_(e_i, (e^p)^{\#}, e_k, (e^q)^{\#}) = R_{pq} R((e^p)^{\#}, e_i, (e^q)^{\#}, e_k)$$