This is something that's been bugging me for a while now, I understand that one of the most important facts about differential geometry is that it doesn't require embeddings into higher dimensional spaces. My problem is how we are then able to meaningfully define coordinates without making reference to an external coordinate system in which our manifold is embedded.
For example, when we define a coordinate chart on the circle, we can use the projection onto the axes as: $$(x,y)\mapsto x\quad (x,y)\mapsto y \tag{1}.$$ This basically takes a circle centred at the origin and provides a four separate coordinate charts which cover the circle. This is fine, but it seems to require us to specify the $(x,y)$ coordinates in the first place. I've seen a similar construction where we begin with the polar angle $\theta$ and use this to define the coordinates in a similar way. But this again seems to require us to have the circle originally embedded in $\Bbb R^2$?
How can we provide coordinates for a simple manifold like the circle without referencing an embedding?
Best Answer
The issue here is that a circle is most naturally a subspace of $\mathbb{R}^n$, usually $\mathbb{R}^2$.
But we can choose other ways to think about a circle that don't describe it as an embedded submanifold of $\mathbb{R}^n$. For example, $[0,1]$ with the points $0$ and $1$ identified is a circle. The quotient $\mathbb{R}/\mathbb{Z}$ is also a circle. Each description of a circle has its uses, but we usually choose the embedding into $\mathbb{R}^2$ for visualization purposes.
The moral is that some manifolds in some contexts most naturally present as an embedded submanifold of $\mathbb{R}^n$, but they also exist as abstract manifolds (a topological space plus smoothly compatible charts).