Geofleur
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In Hassani's Mathematical Physics, a principal fiber bundle is defined as shown below.
I wanted to see if there is a way to view a tangent bundle as a PFB, even if the resulting structure would have to be globally trivial, so I came up with this idea:
Let ##P = {\rm I\!R} \times {\rm l\!R}##, ##G## be the additive group ##(
{\rm I\!R},+)##, and for ##(a,b) \in P## define the action of ##g## as ##(a,b)g = (a,b+g)##, i.e., ##G## consists of translations of the real line. Then a typical element of ##P/G## is ##[(a,0)] = \{(a,r)r\in {\rm l\!R}\}##. Let ## \pi((a,b)) = [(a,0)] ##, and let the map ##s_u## be defined by ##s_u((a,b)) = b##. Then we have ##s_u(pg)=s_u((a,b)g)=s_u((a,b+g))=b+g##, while ##s_u(p)g = s_u((a,b))g = bg = b+g##.
Does this structure count as a PFB, albeit one that can only ever be trivial?
I wanted to see if there is a way to view a tangent bundle as a PFB, even if the resulting structure would have to be globally trivial, so I came up with this idea:
Let ##P = {\rm I\!R} \times {\rm l\!R}##, ##G## be the additive group ##(
{\rm I\!R},+)##, and for ##(a,b) \in P## define the action of ##g## as ##(a,b)g = (a,b+g)##, i.e., ##G## consists of translations of the real line. Then a typical element of ##P/G## is ##[(a,0)] = \{(a,r)r\in {\rm l\!R}\}##. Let ## \pi((a,b)) = [(a,0)] ##, and let the map ##s_u## be defined by ##s_u((a,b)) = b##. Then we have ##s_u(pg)=s_u((a,b)g)=s_u((a,b+g))=b+g##, while ##s_u(p)g = s_u((a,b))g = bg = b+g##.
Does this structure count as a PFB, albeit one that can only ever be trivial?
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