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A smooth $k$-dimensional manifold $M \subset \mathbb{R}^{n}$ is a subset of $\mathbb{R}^{n}$ that can locally be characterized as the graph of some smooth function $\mathbb{R}^{k} \to \mathbb{R}^{n-k}$. In this sense, it makes sense to call this manifold $k$-dimensional since, locally, we only need $k$ out of the $n$ variables to "generate" the remaining variables. This is the technical definition of a manifold.

Intuitively, a manifold to me is a lower dimensional structure in a higher dimensional space. For instance, you can take a line in $\mathbb{R}$, take it into $\mathbb{R}^{2}$, and get an additional direction for you to "move" the line in (which wasn't available to you in one dimension). Similarly, which can talk about two dimensional subsets becoming surfaces in $\mathbb{R}^{3}$ (where we can again move them in an additional direction now), and so forth.

How does one connect the two notions of manifold above? How is the intuition of the fact that a $k$-dimensional manifold in $\mathbb{R}^{n}$ can locally be defined by only $k$ variables that we're taking a $k$-dimensional structure and being able to move it in $n$ directions (instead of just $k$ directions)?

gtoques
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    Just a small comment: Your "manifold" should maybe be called a "submanifold". A manifold is something more abstract in the sense that you don't have a fixed embedding into some $\mathbb R^n$, while a submanifold also has a fixed embedding. For example you can consider a line in 2-dimensional space, but you can also consider lines in 3-dimenional space or arbitrary higher dimension. –  Mar 19 '20 at 10:00
  • @gtoques Here's a way to see the connection: we can generate a "direction of movement" within the manifold by changing one of the $k$ input variables to our smooth function, and doing so for all $k$ variables produces $k$ independent directions of movement within the manifold, which corresponds to the fact that the manifold is $k$-dimensional. – Ben Grossmann Mar 19 '20 at 10:03
  • @PaulK A point that you probably should have made explicit there: no matter which space you embed the line into, every line is the same "manifold" in the usual sense of the term. – Ben Grossmann Mar 19 '20 at 10:06
  • @gt Your intuition of only being able to "move the manifold" in an extra $n-k$ directions is problematic. For instance, the circle is a $1$-dimensional manifold, but it makes perfect sense to move the circle in any direction, i.e. $2$ (as opposed to $2-1 = 1$) dimensions of directions. If you refine this idea, you might get a reasonable interpretation of the cotangent space of a manifold. – Ben Grossmann Mar 19 '20 at 10:10
  • @Omnomnomnom I perhaps didn't articulate the "moving" idea well enough. I meant that a 1-dimensional manifold embedded in a 2-dimensional space (like a circle) can be moved in one additional direction, i.e. a total of 2 directions. If you take a line in $\mathbb{R}$, you can only move it along the real line, but if you take this one dimensional structure into two dimensions, you can now "loop" this line by moving it in one additional direction now (so a total of 2) to make it a circle. – gtoques Mar 19 '20 at 10:16
  • Related (that was my very first question, almost 10 years ago!). – Giuseppe Negro Mar 19 '20 at 10:51

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