How would you define a mathematical space?

[TheDS1337]

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 $R^n$ ? is M just a subset of $R^n$?
Is space a structure (I.E. a set with certain configuration like Wikipedia page says) or is it not a set at all? 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...

 

[fresh_42]

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.

 

[member 587159]

A manifold ##M## is not necessarily a subset of ##\mathbb{R}^n##. In the first place, a manifold is a topological space.

It is something that looks locally like ##\mathbb{R}^n## in a very precise sense, i.e. every point of ##M## has an open neighborhood ##V## together with a homeomorphism ##\phi: V \to U## where ##U## is an open subset of ##\mathbb{R}^n##.

While it is true that many manifolds can be realized as subsets of some Euclidean space ##\mathbb{R}^n## (i.e. we can find a diffeomorphism of our manifold with a manifold that lives embedded in some ##\mathbb{R}^n##), this is not always the case.

 

[TheDS1337]

Very interesting stuff from the both of you, I see now why manifolds and euclidean spaces relate to each other locally.

As you can see, I'm just a newbie into these stuff, I never done topology nor differential geometry before, and definitely not much in abstract algebra.

I need a little help here, asking one more question, I've just started reading "Introduction to Manifolds" by Loring W. Tu, do you guys think it's a good start or if not, can I get some recommendations for a newbie like me from your part? Thanks a lot!

 

[PeroK]

Very interesting stuff from the both of you, I see now why manifolds and euclidean spaces relate to each other locally.

As you can see, I'm just a newbie into these stuff, I never done topology nor differential geometry before, and definitely not much in abstract algebra.

I need a little help here, asking one more question, I've just started reading "Introduction to Manifolds" by Loring W. Tu, do you guys think it's a good start or if not, can I get some recommendations for a newbie like me from your part? Thanks a lot!

What is your level of mathematics currently? That looks very like a graduate text to me.

 

[fresh_42]

Recommending a textbook is in the end always a matter of taste and the individual way people like to learn. I find this one
https://www.amazon.com/dp/0387903577/?tag=pfamazon01-20
a good approach without overwhelming mathematical overhead. But you can easily find lecture notes on the internet, which is cheaper and of less risk in case a book doesn't match your needs.

 

[TheDS1337]

What is your level of mathematics currently? That looks very like a graduate text to me.

Well, I'm actually a graduate student in Physics, I've always been thought from a physicists point of vue "only physics matter", not only that, even the modules I had didn't really require much of mathematics... like the most abstract stuff we did in Relativistic Quantum Mechanics was group theory.

So I wanted to start my mathematical formation because I know I will need it in the future especially that I'm interested in the theoretical part of physics.

 

[PeroK]

Well, I'm actually a graduate student in Physics, I've always been thought from a physicists point of vue "only physics matter", not only that, even the modules I had didn't really require much of mathematics... like the most abstract stuff we did in Relativistic Quantum Mechanics was group theory.

So I wanted to start my mathematical formation because I know I will need it in the future especially that I'm interested in the theoretical part of physics.

What about this?

https://uchicago.app.box.com/s/vabknygqmfkzngv44ru2st30ehpa5ozi

 

[TheDS1337]

What about this?

https://uchicago.app.box.com/s/vabknygqmfkzngv44ru2st30ehpa5ozi

Nice, I'll give it a look!

and ahh... I always find myself asking more questions when talking to mathematicians, I know that a lot of you guys had different paths in learning mathematics, I don't want to create another thread just for this because the question may seem silly... However, How does the average mathematician study different areas of mathematics, and how does one help you with another?

 

[Nikoleta987]

Hello, the topic reminded me of a funny joke :D Let me share it with you!
https://www.jokestjokes.com/Cattle_Fencing_Solutions
Cheers!

 

[member 587159]

I need a little help here, asking one more question, I've just started reading "Introduction to Manifolds" by Loring W. Tu, do you guys think it's a good start or if not, can I get some recommendations for a newbie like me from your part? Thanks a lot!

Prerequisites of this book are the following: Basic knowledge of point set topology (really not too much) and a firm grasp of multivariable analysis, in particular the inverse function theorem is very important.

I used this book together with Lee's book on smooth manifolds in a differential geometry course. I think Tu's book is a good starting point (Lee's book is more advanced) for learning differential geometry.

 

[TheDS1337]

Prerequisites of this book are the following: Basic knowledge of point set topology (really not too much) and a firm grasp of multivariable analysis, in particular the inverse function theorem is very important.

I used this book together with Lee's book on smooth manifolds in a differential geometry course. I think Tu's book is a good starting point (Lee's book is more advanced) for learning differential geometry.

Neat, I've done some point set theory in the past but not so much, I think the hardest part in it are the proofs.