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After reading some introductory texts i've got a question on smooth fibre bundles. I understand that we require both the base space and the typical fibre to be smooth manifolds. But then i am not completely sure why the total space becomes a smooth manifold itself.

Consider a fibre bundle $(E, B, \pi, F, G)$. Then there exist smooth trivializations $\varphi_i:E\rightarrow U_i\times F$ where $U_i$ is open in B. Does the total space E then become a smooth manifold because we can compose these trivializations with those of the base space B and the typical fibre $F$ getting a trivialization $\psi_i:E\rightarrow\mathbb{R}^k\times\mathbb{R}^n\cong\mathbb{R}^{k+n}$ where $k$ is the dimension of B and $n$ is the dimension of $F$?

NDewolf
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    Yes, something along those lines.. – Berci Jun 03 '17 at 22:44
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    Being a smooth manifold is a local property (for reasonable spaces), and fiber bundles are locally products. – anomaly Jun 03 '17 at 22:47
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    Hint: you need to find a system of charts on $E$. The data you are given to get this comprises the systems of charts on $B$ and $F$ and the local product structure from the fibre bundle (what you describe as the trivializations $\phi_i$). Use the local product structure to refine the charts on $B$ and then find charts on $E$ that are products of (refined) charts on $B$ and the given charts on $F$. – Rob Arthan Jun 03 '17 at 22:50
  • But why do we need to refine the charts on B? If we use the product of the charts on B and F, isn't this enough? Cause the product of those open sets in $\mathbb{R}^{k+n}$ is open due to the induced product topology and composition of the trivialization (which is diffeomorphic) and the chart maps is also diffeomorphic. So this should suffice as a chart on E – NDewolf Jun 04 '17 at 10:56
  • I think you are misunderstanding how the local product structure works. The "smooth trivializations" you refer to identify a subset, $E_i$, say of $E$ with a product $U_i \times F$ where $U_i$ is open in $B$. You have to relate these open sets $U_i$ with the given system of charts on $B$. – Rob Arthan Jun 05 '17 at 22:28
  • I did understand the local behaviour. But i didnt really see why i had to refine the charts on B. It is because we need every open set $U_i$ (from the product structure) to be part of a distinct chart? – NDewolf Jun 06 '17 at 08:10
  • No you don't need every open set $U_i$ to be part of a distinct chart on $E$, If you write down the details of the proof, you will find your data comprises two collections of open subsets of $B$, one coming from the manifold structure and one coming from the local product structure. In the proof you will have to relate these. I suggest you try to write down a more detailed proof and ask again if you have further questions. – Rob Arthan Jun 06 '17 at 20:21
  • I have also edited/corrected my question, as it seemed I was talking about using charts on the Lie group $G$ instead of those on $F$. – NDewolf Jul 23 '17 at 13:40
  • I have thought about your answer for a while now and I have been trying to complete the proof. However I am still not sure about it. Is the refinement of the charts on B necessary such that when we try to construct a chart for the local product $U_i\times F$ by pasting together refined charts to cover $U_i$ there is no overlap? I.e. that the new chart covering $U_i$ is a disjoint union of refined charts over B because otherwise the map would not be a homeomorphism? – NDewolf Jul 23 '17 at 15:20

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