I'm trying to learn how to think about principal bundles where the fibre is a lie group with local trivialization [itex]ϕ^{-1}_i:π(U_i)→U_i×G[/itex] . For example [itex]ϕ^{-1}_i:π(S^2)→S^2×U(1)[/itex] (if that makes sense) . But I don't know how to think of this (and other products with lie groups like that) geometrically in 3D space. How does [itex]S^2×U(1)[/itex] look like? I know that U(1) rotates things, but I can't visualize a thing that rotates other things. What does it mean if I put U(1) at every point on [itex]S^2[/itex]? It get's even worse if I want to think about a connection on a principal bundle which is defined on the tangent space of the lie group. I guess there is a lot of higher dimensional stuff going on, but isn't there an easy 3D example which captures every concept at once?(adsbygoogle = window.adsbygoogle || []).push({});

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# Visualizing principal bundles

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