So it makes sense that an inclusion map $$\iota : S \longrightarrow S \subset M $$ maps $$ p \mapsto p $$ But how do you construct these guys in the context of manifolds?
To my understanding, if $S$ is a $k$-dimensional manifold, and we want to explore some property of it using the techniques from undergraduate analysis, then we first take $k$ parameters we can use to define our position $p$ on $S$, then we can construct a map into a space $\mathbb{R}^n$, where $k\leq n$ and check its properties there.
Going by this logic, does that mean that the following are inclusion maps
The inclusion map from the k-dimensional manifold $\mathbb{R}^k$ to $\mathbb{R}^n$ is $$\iota : \mathbb{R}^k \longrightarrow \mathbb{R}^n$$ $$\iota(p_1,\ldots,p_n) = (p_1,\ldots,p_n,0,\ldots,0)$$ (As shown in a theorem).
The unit circle $S^1$ is a 1-dimensional manifold that can be mapped into $\mathbb{R}^2$ by the inclusion $$\iota : S^1 \longrightarrow \mathbb{R}^2$$ by parameter $\theta \in [0,2\pi]$ as $$\iota(\theta)=(\cos(\theta),\sin(\theta))$$
The Torus $T^2$ is a 2-dimensional manifold that can be mapped into $\mathbb{R}^3$ by the inclusion $$\iota:T^2 \longrightarrow \mathbb{R}^3$$ via two parameters $\theta,\phi$ as $$\iota(\theta,\phi)=((R+r\cos\theta)\cos\phi,(R+r\cos\theta)\sin\phi,r\sin\theta)$$
The Torus $T^2$ is a 2-dimensional manifold that can be mapped into $\mathbb{R}^4$ by the inclusion $$\iota:T^2 \longrightarrow \mathbb{R}^4$$ via two parameters $\theta,\phi$ as $$\iota(\theta,\phi)=(\cos\theta,\sin\theta,\cos\phi,\sin\phi)$$
The unit 2-sphere $S^2$ is a 2 dimensional manifold with inclusion map $$\iota : S^2 \longrightarrow \mathbb{R}^3$$ with 2 parameters $\theta, \phi$ as $$\iota(\theta,\phi)=(\cos\theta\cos\phi,\cos\theta\sin\phi,\sin\theta)$$ And so on.
In other words, are all such inclusion maps just the parameterizations of the manifolds we are talking about, as submanifolds of Euclidean space?
I'm interested so I can understand how to explicitly calculate the pullback of forms using inclusion maps to study their properties, but truthfully am unsure If we can just parameterize the manifold as a submanifold everytime or not.
Second question, can inclusion maps ever be charts of manifolds themselves? My initials thoughts are no, because for a chart on a manifold of dimension $n$, you need to be able to map to $\mathbb{R}^n$ and invert continuously. But with the examples of inclusions listed (assuming they are the inclusions), none of them have the same dimension as the number of parameters needed to embed the manifolds.
Any guidance is sincerely appreciated!!