Timeline for Motivation for construction of associated fiber bundle from a principal bundle
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Jan 18, 2019 at 18:17 | comment | added | Praphulla Koushik | At least in case of Lie groups it is more straight forward,,, you just have to consider transition functions $\psi_{\alpha\beta}:U_{\alpha\beta}\rightarrow G$ for $P(M,G)$ and consider the composition $\theta\circ\psi_{\alpha\beta}:U_{\alpha\beta}\rightarrow G\rightarrow H$.. This would anyways give a principal $H$ bundle.. I think it is a good exercise to see that the $H$ bundle we get in this way is same as the one that we get from thinking of $G$ action on $H$ and considering the associated bundle thing... Thanks... I did not observe this at that time... Thanks.. | |
| Jan 18, 2019 at 18:15 | comment | added | Praphulla Koushik | I was re reading this post.. You said "you get a principal $H$-bundle, often denoted $P \times^G H$. The transition maps are clearly just given by applying $\phi$ to those of $P$".. Given a morphism of Lie groups $\theta:G\rightarrow H$, I was thinking this as an action of $G$ on $H$ by $g.h:=\theta(g).h$ then applying the notion of "associated fibre bundle with fibre $F$ for a principal $G$ bundle $P(M,G)$ given an action of $G$ on $F$"... | |
| Sep 29, 2018 at 17:59 | comment | added | Praphulla Koushik | Thanks for the answer.. it did not answer my question that I asked above but it said why association of fiber bundle construction is important.. thanks again... bounty is for that .. | |
| Sep 29, 2018 at 17:51 | history | bounty ended | Praphulla Koushik | ||
| Sep 28, 2018 at 17:20 | vote | accept | Praphulla Koushik | ||
| Sep 28, 2018 at 17:20 | |||||
| Sep 27, 2018 at 5:10 | comment | added | Michael Murray | Ah sorry I just repeated what @DeaneYang has in his answer. | |
| Sep 27, 2018 at 1:31 | comment | added | Praphulla Koushik | I think I have conveyed wrongly what I am wanting to say... I understand that all those bundles arise naturally as associated bundles of one Principal bundle on manifold i.e., the frame bundle.. what I am not seeing is the motivation for that particular construction of fiber bundle from a Principal bundle... @PaulSiegel | |
| Sep 26, 2018 at 19:41 | comment | added | Paul Siegel | @PraphullaKoushik If you find the second paragraph unintelligible then you are learning the subject in the wrong order, and Kobayashi-Nomizu is not the book you should be reading. It assumes (as Ben McKay did) that you are already familiar with various examples of vector bundles on manifolds and why they are important - with that understanding, Ben's answer would tell you that many of the examples and structures that you are already familiar with are unified and clarified by the associated bundle construction. As an alternative, you might start with Marsden-Tornhave's book, for instance. | |
| Sep 26, 2018 at 9:50 | comment | added | Praphulla Koushik | I understand your first paragraph... second paragraph is completely some other foreign language for me.. :( I understand almost nothing there.. please consider using some more words in explaining... | |
| Sep 26, 2018 at 9:47 | history | edited | Ben McKay | CC BY-SA 4.0 |
added answer to part of question
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| Sep 26, 2018 at 9:38 | history | answered | Ben McKay | CC BY-SA 4.0 |