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I am a physicist trying to understand the notion of holonomy in principal bundles.

I am reading about the horizontal lift of a curve in the base manifold of a principal bundle (or just fiber bundle) to the total space and would like to relate it to the "classic" parallel transport one comes across in Riemannian geometry. (I am guessing that the horizontal lift picture of the fiber bundle must be a generalization of the "classic" parallel transport of Riemannian geometry).

In particular, I would like to consider the example of parallel transporting a tangent vector (

To extend this picture in terms of the horizontal lift, I am considering the total space

I am not even sure how, with this approach, after completing a loop one will "move along the fiber"? And how does one eventually relate all this to the picture of moving the vector in ℝ

Any help would be greatly appreciated. Thank you!

I am reading about the horizontal lift of a curve in the base manifold of a principal bundle (or just fiber bundle) to the total space and would like to relate it to the "classic" parallel transport one comes across in Riemannian geometry. (I am guessing that the horizontal lift picture of the fiber bundle must be a generalization of the "classic" parallel transport of Riemannian geometry).

In particular, I would like to consider the example of parallel transporting a tangent vector (

*v*) of*S*(embedded in ℝ^{2}^{3}) in terms of horizontal lift. The picture I have for the "classic" parallel transport (in tangent bundle) is that one transports (infinitesimally) a given tangent vector of*S*as if it were in ℝ^{2}^{3}(*Tℝ*) and then projects it back to the tangent space of^{3}*S*.^{2}To extend this picture in terms of the horizontal lift, I am considering the total space

*E*as*TS*, base manifold^{2}*B*as*S*and the fiber^{2}*E*at_{x}*x*as*T*. The projection map then is clearly_{x}S^{2}*[itex]\pi[/itex]: TS*. However, when trying to find the vertical space of the^{2}[itex]\rightarrow[/itex] S^{2}: (x,v) [itex]\rightarrow[/itex] x*T*, I am getting confused about_{x}E*d[itex]\pi[/itex]:T(TS*. How do I make sense of^{2})[itex]\rightarrow[/itex]TS^{2}*T(TS*in order to find^{2})*ker(d[itex]\pi[/itex])*?I am not even sure how, with this approach, after completing a loop one will "move along the fiber"? And how does one eventually relate all this to the picture of moving the vector in ℝ

^{3}and projecting back to tangent space of*S*.^{2}Any help would be greatly appreciated. Thank you!

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