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Given a fiber bundle $\pi:E\to B$ with fiber $F$ and structure group $G$, we can construct the associated bundle with fiber $\hat{F}$ by using an open cover of trivializations $\{\pi^{-1}(U_\alpha)\}$ and the associated transition functions $\phi_{\alpha \beta}:U_\alpha\cap U_\beta\to G$. Choosing $\hat{F}=G$ gives the associated principal $G$-bundle $E'$.

Now, it seems that to get $E$ back from $E'$, we could do the same thing:

  1. use an open cover of trivializations of $E'$ and their transition functions to change the fiber back to $F$.

Yet it seems that most texts don't do this; instead they

  1. start with $E'\times F$, define an equivalence relation $(p,f)\sim (p\cdot g,g^{-1}\cdot f)$, and define the associated fiber bundle with fiber $F$ to be the quotient $E'\times F / \sim$.

My question is, are 1. and 2. equivalent? If not, why not; if so, why do authors use the seemingly more complicated 2.?

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    They are equivalent and books often show that (or at least state it). The second construction is more "coordinate free" in the sense that the object $E' \times_{\rho} F$ constructed only depends on the global structure of the objects involved (that is, the smooth right action of $G$ on $E'$ and the smooth left action $\rho$ of $G$ on $F$). This helps when you want to transfer things (such as connections, parallel transport, etc) from the principal bundle to the associate bundle without checking every time that everything doesn't depend on the trivializations. – 2017-01-05
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    Do you know where I can find a proof? I'm using Walschap. – 2017-01-05
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    Actually I suppose all this is obvious from the fiber bundle construction theorem, since they have the same transition functions. – 2017-01-05

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