TheDS1337 said:
So, I'm a little confused and I thought I might get some help here.
I have just started learning about manifolds and its super confusing because I've always worked with Euclidean spaces, too much that I didn't even realize it's euclidean and that it has different properties from others.
So my question is, what is space truly?
Why do we say manifold M instead of just R^n ?
Because they are two different concepts. The shortest distance between two points of a manifold is in general not a straight line.
is M just a subset of R^n?
This is a possibility, but the idea behind a manifold is, that we do not have such an embedding. E.g. (a bit exaggerated to demonstrate the point) we both live on the surface of the Earth and there is no way for us to go outside of it, nor inside. The sphere is all that's there: curved, locally flat, and no thought about an outside.
Is space a structure (I.E. a set with certain configuration like Wikipedia page says) or is it not a set at all?
There isn't a mathematical term called "space". We call certain sets in certain settings a space for conveniences. A space in linear algebra is typically a vector space, but it might also be an affine space, spaces in stochastics are phase spaces of possible states, a space in differential geometry does not exist (too ambiguous), a space in algebraic geometry is a variety. Space is a physical term, and even there it needs further explanation for what is meant.
If it's not a set, then why do we even say M is a manifold? what does the letter M here refer to in the first place...
It refers to the fact that it is not flat, not Euclidean. It is an object defined by nonlinear equations, or topological properties. The clue is to forget about embeddings. E.g. you cannot embed the Klein bottle in 3D Euclidean space. The idea behind it is the following:
1. Given a manifold, a set of points which form an "object".
2. O.k., but then we cannot do calculus, so what is it good for?
3. Right. We need one more property: assume it can locally be modeled by a piece of ##\mathbb{R}^n##.
4. How does this help?
5. Continuity and differentiability, as well as convergence are all local properties! Calculus is a theory which only makes statements about the behavior of functions around certain points, a neighborhood of a point. So our requirement allows us to leave the manifold and enter the piece of ##\mathbb{R}^n##, do the calculations, and return the result to the manifold. If our neighborhood is small enough, we will get good results about the manifold.
6. And the costs are?
7. The costs are the fact, that we have different pieces of ##\mathbb{R}^n## at different points. We cannot compare calculations done on one with those done on the other. We won't get rid of locality (unless we assume further properties).
The pieces of ##\mathbb{R}^n## are called charts. They allow us to apply calculus. But they do not replace the manifold, which is still curved. The manifold might be embedded in a bigger ##\mathbb{R}^N##, but we do not care, i.e. we do not require it. It's an other people's problem. All we require is the existence of these charts. They allow us to do calculus on "curved spaces" which were formerly out of reach without an embedding.
