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I'm studying Nakahara's book, Geometry, Topology and Physics and now studying the connection theory. I come across a problem. Please look at the two attachments.

In the attachment , Nakahara said we could use the similar method in the attachment to get [itex]\tilde X[/itex], but why does the first term have [itex]g_i(t)^{-1}[/itex]. According to the first figure, the first term should have the following form

[itex]R_{g_i(t)*}\sigma_{i*}X[/itex]

Since [itex]R_{g*}X=Xg[/itex], it becomes

[itex](\sigma_{i*}X)g_i(t)[/itex]

So there shouldn't be [itex]g_i(t)^{-1}[/itex]. But why did the author put it there? Thank you!

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# Horizontal lift, or parallel transport

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