Short question about principal bundle

[Korybut]

TL;DR

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

 

[gravitation]

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

Which book is this?

 

[Korybut]

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

Which book is this?

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

 

[fresh_42]

Maybe an example can help. Here is one with explicit coordinates and mappings:
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/#4-Hopf-Fibration

$\operatorname{SU}(2)$ as $\operatorname{U}(1)$ principle bundle (Hopf bundle).

 

[martinbn]

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

The point is that the action of $G$ on $P$ preserves the fibers. For each $m\in M$, for each $p\in \pi^{-1}(m)$ and every $g\in G$ you have that $pg\in \pi^{-1}(m)$.

 

[jbergman]

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

The group action respects fibers. In other words the codomain of $\Psi$ restricted to $W\times G$ is $W\times G$. More specifically if we look at a fiber above a single point then $\Psi$ restricted to $p \times G$ codomain is still above p, or $p \times G$.

In plain english, their is a fiber above each point that the group acts on. And the action stays within that fiber. Another way to think about it is that each fiber is isomorphic to the group.

 

[Korybut]

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.

 

[fresh_42]

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.

My summary was:

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.

https://www.physicsforums.com/insights/pantheon-derivatives-part-iii/#Sections

The group action is a permutation of each fiber: transitive, and free.

 

[martinbn]

...
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.

Just to repeat what was already said by every one. You have that $\pi^{-1}(W)$ is the union of the fibers $\pi^{-1}(m)$ over $m\in W$ and each of these fibers is preserved by the action. So the whole $\pi^{-1}(W)$ is also preserved i.e. $\Psi_g(\pi^{-1}(W)) = \pi^{-1}(W)$.

 

[Korybut]

Thanks once again to everyone the notion of principal bundle is perfectly clear to me now