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 (v) of S2 (embedded in ℝ3) 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 S2 as if it were in ℝ3 (Tℝ3) and then projects it back to the tangent space of S2. To extend this picture in terms of the horizontal lift, I am considering the total space E as TS2, base manifold B as S2 and the fiber Ex at x as TxS2. The projection map then is clearly [itex]\pi[/itex]: TS2 [itex]\rightarrow[/itex] S2: (x,v) [itex]\rightarrow[/itex] x. However, when trying to find the vertical space of the TxE, I am getting confused about d[itex]\pi[/itex]:T(TS2)[itex]\rightarrow[/itex]TS2. How do I make sense of T(TS2) in order to find 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 ℝ3 and projecting back to tangent space of S2. Any help would be greatly appreciated. Thank you!