Definition talk:Smooth Vector Bundle

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. &mdash; 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)