Understanding Manifolds: Questions & Answers

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In summary, the conversation discusses the concept of manifolds and their boundaries. It is explained that a manifold without a boundary means that it is continuous and does not have any "kinks" or breaks. They also discuss the purpose of using maps and charts to make calculations on manifolds easier, as well as the importance of these maps being smooth. The use of local homeomorphisms is also mentioned, which means that a manifold is locally like Euclidean space. The definition of a boundary is also mentioned as being a "kink" in the manifold. It is suggested to read the definition first and ask more specific questions for a better understanding.
  • #1
Fellowroot
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I'm just trying to understand something about manifolds.

What is meant when a manifold doesn't have boundary? I thought the boundary was where the manifold "ends" so to speak. Like a boundary point, something where you take a small nhbd (neighborhood) and you get something inside the set and then something outside the set.

But then there are objects that arn't manifolds apparently but they do have boundary.

Another thing. Excuse the simplicity of this question, but why are we so concerned about making maps and charts and atlases and such. What is the whole purpose of doing this? And why do we care about these maps being smooth?

Also what is the purpose of restricting things like here with local diffeomorphisms.

http://en.wikipedia.org/wiki/Local_diffeomorphism

As seen in the link they restrict F which is the map to U. Why do this? What exactly do they mean by this?

Thanks.
 
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  • #2
Fellowroot said:
Another thing. Excuse the simplicity of this question, but why are we so concerned about making maps and charts and atlases and such. What is the whole purpose of doing this? And why do we care about these maps being smooth?Thanks.

One of the reasons charts are used is because you can do calculus on R^n, and so by using the charts you can do calculus on the manifold.

With the second question, do you mean to ask why the charts have to be compatible?
 
  • #3
The reason why you use local homeos. is because a manifold is a space that is not (globally*) quite like Euclidean space, but
it is locally like Euclidean space. And this statement means in a more rigorous way that every point has a neighborhood that is
homeomorphic to R^n , since homeomorphisms preserve basic topological properties of a space. Hope this is what you were asking.

* Not necessarily so, but sometimes so.
 
  • #4
Have you read the definition of the boundary of a manifold?
 
  • #5
A boundary is just a sort of "kink" in the manifold, so that near the boundary point the manifold is not locally like R^n. But I agree with lavinia; to get more out of an answer, it helps if you read the def first and ask something more specific.
 

Related to Understanding Manifolds: Questions & Answers

1. What is a manifold?

A manifold is a mathematical concept used to describe a space that locally resembles Euclidean space. It is a generalization of the concept of a curved surface in three-dimensional space.

2. Why are manifolds important?

Manifolds are important because they allow us to study complex mathematical objects in a simpler way. They also have many practical applications in physics, engineering, and computer science.

3. How are manifolds different from other mathematical objects?

Unlike other mathematical objects, manifolds have a smooth and continuous structure that allows for differentiation and integration. They also have a well-defined notion of distance and curvature, making them useful for studying geometric properties of spaces.

4. What are some common types of manifolds?

Some common types of manifolds include Euclidean spaces, spheres, tori, and projective spaces. Other examples include differentiable manifolds, complex manifolds, and Riemannian manifolds.

5. How are manifolds used in machine learning?

Manifolds play a crucial role in machine learning, particularly in dimensionality reduction techniques such as principal component analysis and t-distributed stochastic neighbor embedding. They also have applications in deep learning, where they are used to model high-dimensional data.

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