I do understand that smooth manifolds are usually defined using the intrinsic view, thus as a Hausdorff space with some certain properties.
However, I'd like to see manifolds as subsets of Euclidean (or complex) vector spaces.
My question:
Which subsets of $\mathbb{R}^n$ or $\mathbb{C}^n$ are smooth manifolds?
And which smooth manifolds are not subsets of $\mathbb{R}^n$ (or $\mathbb{C}^n$)?
According to the Whitney embedding theorem every smooth manifold can be embedded into some Euclidean space. According to a theorem of Nash this can even be done isometrically, if the manifold carries a Riemannian metric. It is usually hard to tell the minimal dimension of the target space as a function of the dimension of the manifold, but there are universal upper bounds (see the links).
A subset $M$ of Euclidean space is a smooth embedded $n$ dimensional manifold iff it can be locally written as a graph over some $n$-dimensional plane iff it admits a cover of smooth charts $\phi: \mathbb{R}^{n}\rightarrow M \subset\mathbb{R}^{n+k}$iff it can be locally written as the zero set of a smooth function $f: U\subset \mathbb{R}^{n+k}\rightarrow \mathbb{R}^k$, see eg https://en.wikipedia.org/wiki/Submanifold