Definition talk:Smooth Vector Bundle

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--Geometry dude (talk) 08:39, 25 September 2014 (UTC)--Geometry dude (talk) 08:39, 25 September 2014 (UTC)== Smoothness of Addition and Scalar Multiplication ==

This is a technical issue that popped up in my head when reviewing my sources on the topic that are silent on the issue. If we take the model fiber of the smooth manifold to be a smooth manifold that is also a vector space, is it automatic that scalar multiplication and addition are smooth? Clearly it is the case if we just equip the (finite dimensional) vector space with the topology and smooth structure from $\R^n$, which is in a way canonical, but what happens in the general case? After all, there exist also exotic smooth structures for example on $\R^4$. So maybe we need to review the definition and replace the word vector space by smooth vector space, i.e. a (finite dimensional) vector space that is also a manifold and where scalar multiplication and addition are smooth. --Geometry dude (talk) 13:55, 24 September 2014 (UTC)

't Has been too long since I immersed myself in the technical details of manifold theory, so I'm not entirely sure. It's probably best to just try and prove it; if it doesn't seem to work in a canonical fashion, we can reconsider. — Lord_Farin (talk) 17:33, 24 September 2014 (UTC)
Well, the answer to that question seems to be non-trivial. I can think of no good reason, why multiplication and addition should be "compatible" with the smooth structure. --Geometry dude (talk) 20:41, 24 September 2014 (UTC)
I've written it out a bit, and indeed, there is no good reason why these maps would be smooth. So I think it is indeed an implicit condition, but we still have to check whether it is actually used in some standard proofs about smooth vector bundles (my gut says it is, but we ought to make sure). If so, I think we should add it. — Lord_Farin (talk) 06:26, 25 September 2014 (UTC)
There's a categorical argument why smoothness of addition and multiplication is needed. Just apply the canonical covariant functor from the category of topological spaces to the category of smooth manifolds on the subcategory "vector bundle"! --Geometry dude (talk) 08:39, 25 September 2014 (UTC)