I'm self-studying smooth manifolds, and there is some terminology that bothers me a lot.
In a lot of books, or homework questions that I looked, there are statements such as
Prove that $S^1 ⊂ R^2$ is a sub manifold.
or
Prove that $GL(n, \mathbb{R})$, i.e the set of all invertible matrices with real entries, is a smooth manifold.
However, there are two possible questions that can be asked in here;
- Either show that there exists a topology and a differentiable structure on the given set that makes the triple - (Set, Topology, Diff. Structure) - a smooth manifold.
- Show that, for example, $S^1$ is a smooth manifold when the topology and the differentiable structure is induced by the smooth manifold $\mathbb{R}^2$.
If the question is the second one, then by definition, $S^1$ is a smooth manifold that is a subset of $R^2$, hence a sub manifold, so nothing to show. Similarly, are they asking $Gl(n)$ is a smooth manifold considered as a subset of $\mathbb{R}^n$ where topology and the smooth structure coming from $\mathbb{R}^n$ ?
If the question is the first one, that is in general not an easy question, I think, but they ask this as if it is an easy exercise given right after the definition.
Question:
Am I really not understanding in here in terms of what is being asked, or required in terms of what needs to be checked to answer those question ? or is it really the authors are sloppy enough that they don't bother with technicalities, and think these are easy questions ?
I mean I have seen lots of authors who considers intuitive arguments as proofs.