What are the key differences between a manifold and a submanifold?

[bronxman]

Hello,

(Continuing learner: new to all of this: patience requested, please.)

What is the difference between a manifold and a submanifold?

I have read and now been working with each. I understand the technical definitions (I think I do... but I do not have an intrinsic FEELING for these definitions). B

Could someone explain the distinction in words for me?

For example: is a submanifold a stepping stone to clearer definition of a manifold? Why define them? Are they really just the same thing and is the distinction made for pedagogical purposes?

 

[ShayanJ]

A submanifold is itself a manifold. The only point is that its underlying set is the subset of the underlying set of another manifold.
For example a 2-sphere is a manifold by itself. But if you consider a 2-sphere in a 3D Euclidean space, then the 2-sphere is a submanifold of that Euclidean space.

 

[bronxman]

A submanifold is itself a manifold. The only point is that its underlying set is the subset of the underlying set of another manifold.
For example a 2-sphere is a manifold by itself. But if you consider a 2-sphere in a 3D Euclidean space, then the 2-sphere is a submanifold of that Euclidean space.

OK... I get that. I am reading Frankel's Geometry of Physics. And struggling.

So then... why does he define submanifold first?
1.1c: The Main Theorem on Submanifolds
1.2c: A Rigorous Definition of a Manifold

Why define one first. Somewhere here is an answer to my confusion.

In fact,in 1.3c. The Tangent Space to Mⁿ at a point

he goes back to calling Mⁿ a submanifold.

 

[ShayanJ]

OK... I get that. I am reading Frankel's Geometry of Physics. And struggling.

So then... why does he define submanifold first?
1.1c: The Main Theorem on Submanifolds
1.2c: A Rigorous Definition of a Manifold

Why define one first. Somewhere here is an answer to my confusion.

Note that the complete phrase is "The main theorem on submanifolds of $\mathbb R^N$".
Yeah, he starts with submanifolds of $\mathbb R^N$ which may confuse some readers, like you. The only thing is that, he's just assuming people know what it means for a manifold to be a submanifold of $\mathbb R^N$ because of the intuitions like a surface in a 3D Euclidean space. So it seems to me here he's requesting the reader to use his\her intuition. Note that he doesn't talk about submanifolds of other manifolds. For that he needs the formal definition of the manifold first.

 

[bronxman]

Note that the complete phrase is "The main theorem on submanifolds of $\mathbb R^N$".
Yeah, he starts with submanifolds of $\mathbb R^N$ which may confuse some readers, like you. The only thing is that, he's just assuming people know what it means for a manifold to be a submanifold of $\mathbb R^N$ because of the intuitions like a surface in a 3D Euclidean space. So it seems to me here he's requesting the reader to use his\her intuition. Note that he doesn't talk about submanifolds of other manifolds. For that he needs the formal definition of the manifold first.

OH! Thank you!

 

[lavinia]

A submanifold is not only a manifold and a subset of a another manifold but is is a submanifod in the subspace topology. This means that around any point there is an open neighborhood in the larger manifold whose intersection with the sub manifold is homeomorphic (or diffeomorphic) to and open subset of Euclidean space.

Here is an example to think about. Consider the torus to be a square with opposite edges pasted together. Now look at the straight line that make an irrational angle to one of the edges - say the bottom edge. This line is a manifold and is a subset of the torus. Is it a sub manifold?

 

[ShayanJ]

A submanifold is not only a manifold and a subset of a another manifold but is is a submanifod in the subspace topology. This means that around any point there is an open neighborhood in the larger manifold whose intersection with the sub manifold is homeomorphic (or diffeomorphic) to and open subset of Euclidean space.

Let me restate it to see whether I got it or not. At first we have a set and then consider a subset of it. Now I give the bigger set a topology and its subset another topology. And then I give both of them a manifold structure. So you're saying that the subset will result in a submanifold only if the topologies given to the set and the subset are the same?

Here is an example to think about. Consider the torus to be a square with opposite edges pasted together. Now look at the straight line that make an irrational angle to one of the edges - say the bottom edge. This line is a manifold and is a subset of the torus. Is it a sub manifold?

If I'm not mistaken, you're talking about the diagonal of the square. After pasting the first pair of edges and getting a cylinder, the diagonal should become a helix. But I have problem visualizing it further. What I can think of, is that you mean the line should get stretched, right? But how is it related to the above comment? Also the whole surface should get stretched as you transform it from a square to a torus. Can you clarify more?

 

[lavinia]

Let me restate it to see whether I got it or not. At first we have a set and then consider a subset of it. Now I give the bigger set a topology and its subset another topology. And then I give both of them a manifold structure. So you're saying that the subset will result in a submanifold only if the topologies given to the set and the subset are the same?

yes.

If I'm not mistaken, you're talking about the diagonal of the square. After pasting the first pair of edges and getting a cylinder, the diagonal should become a helix. But I have problem visualizing it further. What I can think of, is that you mean the line should get stretched, right? But how is it related to the above comment? Also the whole surface should get stretched as you transform it from a square to a torus. Can you clarify more?

The diagonal of the square is at a 45 degree angle. This is not irrational.

At an irrational angle the line reaches some point at one of the edges then continues on starting at the opposite point on the opposite edge. When it reaches the edge a second time, it again does not stop but continues at the new opposite point. This continues forever since at an irrational angle it will never close off. The remarkable fact is that the entire line is dense in the torus. That is, any point on the torus is arbitrarily close to some point on the line.

 

[micromass]

I guess to distinguish between those, you would need to talk about immersed or embedded submanifolds. But I guess that it makes most sense to take a submanifold as always embedded.

 

[mathwonk]

to repeat what has been said, he starts with 1) manifolds which occur in R^n, then more generally 2) arbitrary manifolds. that's why he thought his choice of ordering of topics was natural, since it went from the more special to the more general. Unfortunately his choice to use the word "submanifold", made it seem the opposite.