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Constructing atlas

  1. Mar 10, 2014 #1
    I'm trying to get a better understanding of some topology for a GR class I'm taking...I'm wondering if someone can help me understand how to go about constructing atlases or just charts in general. I understand the concept but I am trying to get a better handle on the math.
     
  2. jcsd
  3. Mar 11, 2014 #2

    SteamKing

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    Well, that's a pretty broad topic. What do you know about map projections?
     
  4. Mar 11, 2014 #3
    Not a lot...It might help me to see simple examples like the two torus or a sphere...
     
  5. Mar 11, 2014 #4

    SteamKing

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    Well, I don't know of anyone who has constructed an atlas on a torus.

    However, map projections are mathematical tools which transform the surface of a sphere to a flat, two-dimensional representation.

    http://en.wikipedia.org/wiki/Map_projection
     
  6. Mar 11, 2014 #5
    Well, the reason I bring up the two torus is because one of the sample problems in the book I'm using is to "prove that the two torus is a manifold by explicitly constructing an appropriate atlas." Well, I did fail to mention it need not be a maximal one.
     
  7. Mar 11, 2014 #6

    micromass

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    So, how did you define the torus in the first place?
     
  8. Mar 11, 2014 #7
    What do you mean (x^2+y^2+z^2 + R^2 - r^2)^2 = 4R^2(x^2+y^2). Sorry, I'm quite a novice
     
  9. Mar 11, 2014 #8

    SteamKing

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    'Atlas' must mean something else in this context. I assumed you were talking in the OP about a book of maps.
     
  10. Mar 12, 2014 #9

    mathwonk

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    an atlas (on a surface) is a collection of "discs" that cover the space. plus maps from each disc to an ordinary disc in the plane. so just look at a sphere and try to cover it with distorted discs or rectangles, or a torus.

    it id pretty easy to see that a sphere can be covered by two discs, one covering a little more than the northern hemisphere, and one covering a little more than the southern hemisphere. It will take me a little visualizing to think of how many rectangles it takes to cover a torus. One clue is to picture a torus as a rectangle with identifications. then it seem you can easily cover it with 4 rectangles, but i am a little "sleepy".

    i.e. a torus is a union of two cylinders and each cylinder is a union of 2 rectangles, isn't it?

    or just take a handful of paper discs and try to overlap them and form a surface. imagine what you could obtain.
     
  11. Mar 14, 2014 #10

    WWGD

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    You can construct an atlas for a torus S1 x S1 by using atlases for each of the S1 s. If M,N are manifolds, so is MxN.
     
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