Understanding Manifolds: Questions & Answers

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Discussion Overview

The discussion revolves around the concept of manifolds, specifically addressing the nature of boundaries in manifolds, the purpose of charts and atlases, and the significance of local diffeomorphisms. Participants explore these topics from a theoretical perspective, raising questions and providing insights into the mathematical framework of manifolds.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the meaning of a manifold without a boundary, suggesting that boundaries are where a manifold "ends" and expressing confusion about non-manifold objects that possess boundaries.
  • Another participant emphasizes the importance of charts and atlases for performing calculus on manifolds, indicating that these tools facilitate calculations similar to those in Euclidean spaces.
  • A different participant explains that local homeomorphisms are used because manifolds are locally similar to Euclidean space, though not necessarily globally, and that this local similarity is crucial for understanding manifold properties.
  • One participant prompts others to consider the definition of the boundary of a manifold, implying that a clearer understanding may lead to more specific questions.
  • Another participant describes a boundary as a "kink" in the manifold, suggesting that near boundary points, the manifold does not resemble R^n, and encourages reading definitions for deeper insights.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the concepts discussed, with some seeking clarification on definitions and others providing explanations. There is no clear consensus on the interpretations of boundaries or the necessity of certain mathematical structures.

Contextual Notes

Some participants reference definitions and concepts that may require prior knowledge, indicating that the discussion may be limited by assumptions about the participants' familiarity with manifold theory.

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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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?
 
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.
 
Have you read the definition of the boundary of a manifold?
 
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.
 

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