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I was reading the proof of why $(\nabla_XY)=\lim_{t\to 0} \frac{P^\gamma_{0t}Y(\gamma(t))-Y_p}{t}$ and this answer provided a good explanation, although I don't understand why $$ (\nabla_X Y)(c(t)) = \frac{DY(c(t))}{dt} = \dot{Y}^i(t) \xi_i(c(t)) + Y^i(t) \frac{D\xi_i(t)}{dt} = \dot{Y}^i(t) \xi_i (c(t)) $$ In particular this implies $$Y^i(t) \frac{D\xi_i(t)}{dt} = 0$$ which would mean $\xi_i(t)$ are constant functions. We defined them as parallel transport of the basis vectors of $T_pM$ along $\gamma$. But I dont see why are they constant.

Edit: I suppose if we look at $\xi_i(t)$ as a function $t\mapsto P^\gamma_{0t}(\xi_i)=:\xi_i(t)$, where $\xi_i(t)$ is a unique parallel vectorfield along $\gamma$ with $\xi_i(0)=\xi_i$. And to be parallel along $\gamma$ means exactly $\frac{D\xi_i(t)}{dt} = 0$ by definition. Is my reasoning correct?

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Yes, your reasoning is perfectly fine. The covariant derivative along $\gamma$ of the $\xi_i$ vanishes, since the $\xi_i$ are parallel along $\gamma$.

(Would put this as a comment, since you answered the question yourself, but I don't have enough reputation to comment.)

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