Short question about principal bundle

gravitation said:

##\chi## is a local trivialization, but the action ##\Psi## preserves the fibers.

Which book is this?

Korybut said:

Sorry, but I don't get you clarification. How LOCAL trivialization is aware of of the whole manifold ##P## since group might send this neighbourhood ##W## in general to any other domain of ##P##? Book is "Differential Geometry" by Rudolph and Schmidt

Korybut said:

TL;DR Summary: Free Lie group action

Hello there!

Book provides the following definition
Let ##(P,G,\Psi)## be a free Lie group action, let ##M## be a manifold and let ##\pi : P \rightarrow M## be a smooth mapping. The tuple ##(P,G,M,\Psi,\pi)## is called a principal bundle, if for every ##m\in M## there exists a local trivialization at ##m##, that is, there exist an open neighbourhood ##W## of ##m## and a diffeomorphism ##\chi:\pi^{-1}(W)\rightarrow W \times G## such that
1. ##\chi## intertwines ##\Psi## with the G-action on ##W\times G## by translations on the factor ##G##,
2. ##pr_W \circ \chi(p)=\pi(p)## for all ##p\in \pi^{-1}(W)##.

To clarify the notation used ##\Psi## is the map ##\Psi: P\times G\rightarrow P##

My question is about the first property of ##\chi##. I can act with my group using ##\Psi## on ##\pi^{-1}(W)##. Moreover in general I can obtain
## \Psi_g (\pi^{-1}(W))\cap \pi^{-1}(W)=0## (##g\in G##) so ##\chi## has nothing to do with ## \Psi_g (\pi^{-1}(W))##. Here I got puzzled: ##\chi## is supposed to commute with group action however it might not even exist on the corresponding domain. Or this ##\chi## is supposed to be defined on all ##\Psi_g(\pi^{-1}(W))## for all ##g\in G## ?

Thanks in advance

Korybut said:

Thanks to everyone for help. I kinda get this formal definition

I would like to summarize just in case

Manifold ##P## should be designed in the way that each ##\pi^{-1}(W)## is diffeomorphic to ##W\times G##. One can act on the latter with any element of ##G## in the obvious way.
However I find that my first guess was right. This diffeomorphism ##\chi## is actually defined on ##\Psi_g (\pi^{-1}(W))## for all ##g\in G##, all the examples I found so far are of this type. Than there is no trouble with the first property of the definition.

If we have in addition [to a fiber bundle] a continuous operation ##(E , G) \rightarrow E## of a topological group, e.g. a Lie group, on the total space ##E## of a fiber bundle, then ##(E,X,\pi,F,G)## is called a principal bundle, if the group operation maps each fiber ##E_x## on itself, i.e. ##\pi (xg)=\pi(x)## for all ##x \in E\, , \,g\in G##, the group operates freely (only ##g=1 \in G## leaves points in a fiber invariant) and transitive (all points ##y \in E_x## in a fiber can be reached by some ##g\in G##). ##G## is called the structure group of the principal bundle.

Korybut said:

...
However I find that my first guess was right. This diffeomorphism ##\chi## is actually defined on ##\Psi_g (\pi^{-1}(W))## for all ##g\in G##, all the examples I found so far are of this type. Than there is no trouble with the first property of the definition.