Talk:Line bundle

Projective modules

What about line bundles as finitely generated projective modules with constant rank 1? 131.111.28.82 (talk) 15:59, 8 January 2009 (UTC)

You can do that in those cases where line bundles are finitely generated by global sections. For example in algebraic geometry this is only valid under the extra assumption that the base space is affine. In complex analytic geometry it is only valid if the base space is Stein. This is called the Serre-Swan Theorem and I agree that it should be part of the article on vector bundles. Spaetzle (talk) 12:09, 1 March 2016 (UTC)

"The most important"

Saying anything is the "most important" anything is either editorial POV, or possibly "original research". In any case it falls pretty much outside our house style. And probably a topologist and algebraic geometer would disagree on precisely which universal line bundles where are important. Charles Matthews (talk) 13:52, 2 March 2010 (UTC)

OK, I concede that it's a POV statement. Nevertheless I suspect that I could find a reference for it. If you object, go ahead and remove it. I won't revert, because I think I'm showing my own bias; after all, this is my favorite line bundle ever. Ozob (talk) 00:49, 3 March 2010 (UTC)

Definition

I don’t see a definition of ‘line bundle’ here—only examples. NormHardy (talk) 17:34, 30 September 2012 (UTC)

There is a definition: A line bundle is a vector bundle of rank one. However, I don't like the definition the way it is. Right now it only refers to topology and differential geometry. This is quite unnecessary. As far as I see, it is true in every context where vector bundles play a role that a line bundle is defined as a vector bundle of rank one. Spaetzle (talk) 12:09, 1 March 2016 (UTC)

Merging in of line field

It has been proposed to merge line field into line bundle. As the concept of a line bundle is very general and of great importance in many branches of geometry and as line fields seem quite exotic to me, I would rather not merge. Spaetzle (talk) 12:09, 1 March 2016 (UTC)

• Agree, don't merge, the line field seems to be a section of a line bundle, does not indicate whether it needs to be an integrable section of not. If if is integrable, then that article would need to talk about ... well, lots of things. 67.198.37.16 (talk) 14:45, 19 June 2016 (UTC)

I also agree. A line field is a section of the projectivized tangent bundle, so it's never a section of a line bundle! Merging it here would be disastrous. Ozob (talk) 19:55, 19 June 2016 (UTC)

I've removed the merge templates. Ozob (talk) 19:56, 19 June 2016 (UTC)

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