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In the red line part of below picture ,why the $\partial_t(\Gamma_{ip}^k \Gamma_{jl}^p -\Gamma_{jp}^k\Gamma_{il}^p)$ vanish ?I know $\Gamma$ will vanish under normal coordinate. But if so, the RHS of red line will equal to $0$. And there is similar question in $\frac{\partial}{\partial t}\Gamma_{jl}^h$ of the third line above red line .

Thanks for any useful hint or answer. And the below picture is from 183th page of the paper

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The term

$$\tag{1} \partial_t(\Gamma_{ip}^k \Gamma_{jl}^p -\Gamma_{jp}^k\Gamma_{il}^p)$$

is not the same as the term

$$\tag{2} \partial_t \Gamma_{ip}^k .$$

The first one is zero at $x$. Because you have to apply product rule to this term, and after that, all terms will have an zeroth order term $\Gamma_{ij}^k$ there, so they are all zero at $x$ under normal coordinate. However, you don't have a zeroth order term on $(2)$.

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