Constructing Atlases: Understanding Topology for GR Class

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In summary, constructing atlases for a torus is a mathematical tool that transforms a sphere into a 2-dimensional representation. An atlas on a torus is a collection of "discs" that cover the space, and each disc has a map to an ordinary disc in the plane. A sphere can be covered by two discs, while a torus can be covered by four rectangles or two cylinders.
  • #1
black_hole
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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.
 
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  • #2
Well, that's a pretty broad topic. What do you know about map projections?
 
  • #3
Not a lot...It might help me to see simple examples like the two torus or a sphere...
 
  • #4
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
 
  • #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.
 
  • #6
So, how did you define the torus in the first place?
 
  • #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
 
  • #8
black_hole said:
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.
 
  • #9
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.
 
  • #10
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.
 

1. What is topology and why is it important for understanding GR class?

Topology is the branch of mathematics that studies the properties of space that are preserved under continuous deformations, such as stretching or bending. In the context of GR class, topology is important because it helps us understand the structure and connectivity of space-time, which is crucial for understanding the behavior of gravity and predicting the motion of objects.

2. What is an atlas and how is it used in topology?

An atlas is a collection of charts or maps that cover a space and allow us to understand its topological properties. In topology, an atlas is used to define the local coordinates and transformations on a space, which are essential for understanding the global topology and its features.

3. How do we construct an atlas for a given space?

To construct an atlas for a given space, we need to divide the space into smaller regions and define a chart for each region. The charts should overlap and have consistent transformations at the boundaries to ensure a smooth transition between regions. This process can be repeated until the entire space is covered, resulting in a complete atlas.

4. What are the different types of atlases used in topology?

There are two main types of atlases used in topology: smooth and piecewise-linear. A smooth atlas consists of charts that are infinitely differentiable, while a piecewise-linear atlas consists of charts that are only differentiable up to a certain order. The choice of atlas depends on the complexity and smoothness of the space being studied.

5. How does understanding topology help us solve problems in GR class?

Understanding topology helps us solve problems in GR class by providing a framework for analyzing the structure and properties of space-time. By using techniques from topology, we can identify the global topology of a space, which is crucial for making predictions about the behavior of objects and understanding the effects of gravity.

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