@0celo7 No, that's why I put it in quotes, but it sounds like "Reopen my question, or I will continue to get on your nerves". Hm, probably not blackmail. Extortion?
@ChrisWhite Lemma 4.5.6 is satanic. The comment "The first and third terms vanish as $\gamma(t)$ is an unbroken geodesic" is misleading as hell because those aren't the only terms that vanish as a result of that. They just don't bother mentioning the others!
As a result, I spent 30 minutes trying to figure out why $$\frac{\mathrm{D}}{\partial u_2}\frac{\partial}{\partial u_1}=0$$ since that's the part common to the terms they mention. Turns out that's not zero and they vanish because of other reasons.