Thanks for the answer it seems clearer than how I thought about it. So $\Pi_{tX}(Y_p)\approx Y_{\phi^X(t)}-t\nabla_XY|_{\phi^X(t)}+t^2\nabla^2_XY|_{\phi^X(t)}-...$. Ignoring higher order terms one gets $\Pi_{tX}(Y_p) \approx Y_{\phi^X(t)}-t\nabla_XY|_{\phi^X(t)}$

@Qfwfq I replied before continuing reading the thread so I didn't notice, sorry for that. I'll keep my reply though because I think the answer in its link might be useful for any future viewers of this post

@Qfwfq Not really. you can differentiate along the flow in the covariant derivative too. what matters is the way you transport the vectors. lie derivatives don't need a notation of parallel transport. see the answer here: math.stackexchange.com/questions/2481628/…

@Qfwfq I want to know this too. my idea is that it doesn't matter the curve as long as the parallel transport map of the connection is the one used. but I'm not sure.

@MichaelWeiss this is really what got me to like DG. it can get very abstract but the main reasoning behind most of the ideas is still "geometric". which makes it tasteful at least to my liking.

Before seeing this I asked about an approximation of the parallel transport along the flow of a vector field. The result looks similar but I'm not so sure about all the steps. Can you take a look if you're available? mathoverflow.net/questions/398269/…

This IMO is the derivation of RCT that makes the most sense