Concept of Manifold

Question

The concept of manifolds is freaking me out.

For me it seems like a manifold is just a subspace embedded in a higher dimension. In order to clear out my confuision I have created a list and I would be glad if someone could tell me if my guesses are right or not, and why they are wrong.

1. Sphere in $\mathbb R^n$: is a smooth manifold
2. Cube in $\mathbb R^n$: is not a smooth manifold but a topological manifold
3. Line in $\mathbb R^n$: is not a manifold as there are no open neighborhoods for points on the line.
4. Point in $\mathbb R^n$: not a manifold as there are no open neighborhoods.

Answer 1

A sphere is a (2D) manifold that can admit a smooth structure.

A cube, I interpret to mean as $[0,1]^3$ is not even a topological manifold.

A line is a (1D) manifold that can admit a smooth structure.

A point is a (0D) manifold that can admit a smooth structure.

If you are looking for a manifold without a smooth structure, you will have a difficult time. Every manifold in 1,2,and 3 dimensions has a smooth structure.

Edit if by "cube" you mean "boundary of cube" then that is a 2d manifold with a smooth structure (as ALL 2d manifolds have smooth structures).

Answer 2

To clear up some confusion, recall that a $k$-manifold $M$ is a second-countable, Hausdorff space which is locally homeomorphic to $\mathbb{R}^k$. The last requirement is the most important, at least it's the one that can be understood better since it says something about the intrinsic geometry. To make things a bit easier, lets look at all of your examples in the case where $k = 0,1,2,3$.

If you take any point of the sphere and draw an open set around it, it looks bent disk which can be unbent smoothly to get a regular disk that you can place in the plane.

The cube has some nice open sets which look like 3-disks, but if you look at any 3-disk about a vertex (i.e pointed boundary) then you can continuously transform it to a 3-disk.

If you take any point on a line and make a neighborhood about it, it looks like an open interval in the plane.

The topology for a space with one point, it the trivial topology, i.e the point itself is an open set. The definition for manifold requires that for every point, there exists! and open set about it with a certain property. Thus taking the open set to be the point, that just looks like a number on the real line.

In these examples, whatever the neighborhoods on these objects are homeoomorphic to, that determines their dimension. So the sphere is a smooth 2-manifold, the cube is a smooth 3-manifold, the line is a smooth 1-manifold, and the point is a smooth 0-manifold.