Constructing Atlases: Understanding Topology for GR Class

[black_hole]

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.

 

[SteamKing]

Well, that's a pretty broad topic. What do you know about map projections?

 

[black_hole]

Not a lot...It might help me to see simple examples like the two torus or a sphere...

 

[SteamKing]

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

 

[black_hole]

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.

 

[micromass]

So, how did you define the torus in the first place?

 

[black_hole]

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

 

[SteamKing]

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.

'Atlas' must mean something else in this context. I assumed you were talking in the OP about a book of maps.

 

[mathwonk]

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.

 

[WWGD]

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.