Definition talk:Smooth Curve

Is a non-compact connected manifold of dimension 1 by definition identical with the Real Number Line? If so, we need a page to demonstrate this -- and then it may not be necessary to define the domain as being such a manifold, we can just make it the Real Number Line. If they are not necessarily the same object, then we may need to reword this definition. --prime mover (talk) 12:05, 16 September 2014 (UTC)

Well, it's a bit subtle. The definition of smoothness used here is the one for manifolds, in fact it is the one we need. The real line or an interval by itself is not a manifold, just a set. One still needs to put a topology and a smooth structure on it, that is the 'standard topology' and 'standard smooth structure'. As I wanted to circumvent using these as they are not defined, I defined it like this. The underlying proposition is that any smooth, non-compact, connected $1$-manifold is diffeomorphic to the real line equipped with the standard topology and smooth structure. So the two are isomorphic in the smooth category. --Geometry dude (talk) 12:49, 16 September 2014 (UTC)

In my opinion it would be an asset to the site if, in due time, we had this information as a subpage (e.g. "/Manifolds") on the definition of "open real interval", with appropriate links to theorems demonstrating the facts mentioned above. However this still requires considerable preparation (as is often the case) and might not be feasible at this moment in time. Still, I think it's a good thing to strive for. — Lord_Farin (talk) 13:53, 16 September 2014 (UTC)

So if a non-compact connected manifold of dimension 1 is *not* identical to the real number line then it's incorrect to say "i.e. the real number line". You *may* say "e.g. the real number line" and then in "also see" provide a link to prove that the r.n.l is a n-c-cmod1 but IMO that's a distraction. --prime mover (talk) 17:07, 16 September 2014 (UTC)