How To define charts and atlases

[saminator910]

I've been reading on differential topology, and all the examples they give me are very abstract . They speak of arbitrary charts (V,ψ), and (U,φ), which map pieces of the m-manifold onto ℝm, which is fine, I understand the concepts, but how does one describe the functions for real, certainly you don't just say that there is some function named ψ, that maps V to ℝm, you need to include the actual function in some cases. So what exactly are these functions, can someone give me an actual tangible example, preferably on something simple like a 1-manifold? P.S. I am familiar with the stereographic projections for the unit sphere, but I don't know how to reproduce something of similar effect on a different manifold.

 

[tiny-tim]

hi saminator910!

an easy example would be an atlas of the Earth with two charts, one for the whole Earth except for the arctic, and one for the whole Earth except for the anatarctic …

they "agree" where they overlap, but they can't be combined into a single chart

 

[meldraft]

On the practical side, a local coordinate system would be an example of a chart.

 

[saminator910]

thanks, but I think you missed the point of my question. Just as there are stereographic projection functions which chart the sphere minus one point onto ℝm, and you use two to map the entire sphere, does one produce similar things for different manifolds? and if one does, can someone provide an example of one of these functions?

 

[jgens]

Just as there are stereographic projection functions which chart the sphere minus one point onto ℝm, and you use two to map the entire sphere, does one produce similar things for different manifolds?

If you want to show that a particular object is a manifold, then you will have to demonstrate that it has an atlas consisting of overlapping charts or any other equivalent statement. For example:

1. Define the n-dimensional sphere by S^n = \{(x_1,\dots,x_{n+1}) \in \mathbb{R}^{n+1} : |x_1|^2+\cdots+|x_{n+1}|^2 = 1\}. Now consider the open set U_k^{\pm} and the map \phi_k^{\pm}:U_k^{\pm} \rightarrow B^n defined as follows...
   <br /> U_k^{\pm} = \{(x_1,\dots,x_{n+1}) \in S^n : \mathrm{sgn}(x_k) = \pm 1\}<br />
   <br /> \phi_k^{\pm}:(x_1,\dots,x_k,\dots,x_{n+1}) \mapsto (x_1,\dots,\hat{x_k},\dots,x_{n+1})<br />
   where the hat indicates that the entry is omitted. The collection \{(U_k^{\pm},\phi_k^{\pm})\} is an atlas on S^n as can be easily checked.
2. Define the n-dimensional real projective space as \mathbb{R}P^n = S^n / \sim, where x \sim y iff x = \pm y. Let \pi:S^n \rightarrow \mathbb{R}P^n be the natural quotient map and notice that \pi|U_k^+ is a homeomorphism onto its image. Now define the open sets V_k and the maps \psi_k:V_k \rightarrow B^n as follows...
   <br /> V_k = \pi(U_k^+)<br />
   <br /> \psi_k = \phi_k^+ \circ (\pi|U_k^+)^{-1}<br />
   As before, it is easily checked that \{(V_k,\psi_k)\} is a coordinate system on the real projective space.
3. Let \mathrm{U}(n) be the collection of all unitary matrices and let \mathfrak{u}(n) be the collection of all skew-hermitian matrices. It is well-known that the matrix exponential is a homeomorphism onto its image in some neighborhood of 0 and an easy calculation shows that if this neighborhood is chosen sufficiently small, then the matrix exponential maps skew-hermitian matrices onto unitary matrices (in this neighborhood). This gives us an open set W \subseteq \mathrm{U}(n) and an embedding \theta:W \rightarrow \mathfrak{u}(n). The remaining coordinate charts are constructed by composing with left translations.

Each of these constructions furnishes an atlas on the space in question. To show that these spaces are manifolds, we usually want to show that they have a second countable and Hausdorff topology, but this is easy. I chose these three example because they each give a common technique used to construct coordinate systems: the first gives a coordinate system from scratch; the second gives a new coordinate system by using a previous one; and the third uses a trick to produce a coordinate system. As one last note, aside from establishing that something is a manifold, it's usually not important to have a concrete charts in mind. Often this is true even when you are working with a specific manifold like S^n. So try not to worry about it too much.

 

[saminator910]

Thanks, you answered my main question *generally, one does not necessarily need to explicitly define the chart's functions. In your first example the functions map the n-Sphere from ℝn+1 to ℝn by leaving out a coordinate, correct? So on a 2-sphere you would have three of these, each one omitting a single coordinate, "flattening" it onto ℝ2? So, say I wanted to construct a smooth differential structure on an ellipsoid, could you use similar charts for the atlas? and what about a hyperbolic paraboloid?

 

[jgens]

In your first example the functions map the n-Sphere from ℝn+1 to ℝn by leaving out a coordinate, correct? So on a 2-sphere you would have three of these, each one omitting a single coordinate, "flattening" it onto ℝ2?

Yes to the first question. There are two things I want to mention though.

1. The image of each of these maps is the unit ball in \mathbb{R}^n. You can, of course, make the image \mathbb{R}^n by composing with a suitable diffeomorphism of these sets.
2. You actually get 2n+2 charts on the n-sphere using this process. For the kth coordinate there is a positive hemisphere and a negative hemisphere, and the charts are given by flattening these hemispheres out onto the unit ball.

So you just have to be a little bit careful with things here.

So, say I wanted to construct a smooth differential structure on an ellipsoid, could you use similar charts for the atlas? and what about a hyperbolic paraboloid?

I would show that an ellipsoid is the regular level set of a smooth function \mathbb{R}^3 \rightarrow \mathbb{R} rather than constructing charts. This same trick can also show that the n-sphere and hyperbolic paraboloids are manifolds.

 

[lavinia]

I've been reading on differential topology, and all the examples they give me are very abstract . They speak of arbitrary charts (V,ψ), and (U,φ), which map pieces of the m-manifold onto ℝm, which is fine, I understand the concepts, but how does one describe the functions for real, certainly you don't just say that there is some function named ψ, that maps V to ℝm, you need to include the actual function in some cases. So what exactly are these functions, can someone give me an actual tangible example, preferably on something simple like a 1-manifold? P.S. I am familiar with the stereographic projections for the unit sphere, but I don't know how to reproduce something of similar effect on a different manifold.

Anytime you can picture an open disk on a surface that is a picture of a domain that is homeomorphic to an open set in the plane. These are the intuition for all coordinate charts including in higher dimensions. For example one can imagine small polar caps around any point on the sphere. If you actually want to construct the homeomorphisms then take a standard cap at the north pole and your chart is projection onto the xy-plane. For any other points, take a congruent cap around it, rotate it to the north pole then project onto the xy-plane.

A good exercise is to define an atlas on the flat torus.

 

[saminator910]

I guess I'm just confused as to how the mapping should "look" afterward, for example, since one need only prove that a m-manifold is homeomorphic to ℝ^m, so for the hyperbolic paraboloid,
h={(x₁,x₂,x₃)|x₁²-x₂²-x₃=0}
one chart should be able to map that onto ℝ², correct? so could you use that same method of removing the x₃ coordinate? that would literally produce ℝ², but it seems too trivial. but even the fact that the hyperbolic paraboloid is composed of a continuous function from ℝ² to ℝ means that it is a homeomorphism of ℝ², correct?

 

[lavinia]

I guess I'm just confused as to how the mapping should "look" afterward, for example, since one need only prove that a m-manifold is homeomorphic to ℝ^m, so for the hyperbolic paraboloid,
h={(x₁,x₂,x₃)|x₁²-x₂²-x₃=0}
one chart should be able to map that onto ℝ², correct? so could you use that same method of removing the x₃ coordinate? that would literally produce ℝ², but it seems too trivial. but even the fact that the hyperbolic paraboloid is composed of a continuous function from ℝ² to ℝ means that it is a homeomorphism of ℝ², correct?

Yes for the hyperbolic paraboloid just project onto the xy-plane.
Generally though a chart will not equal all of Euclidean space but only an open set within it.

Note that whenever you graph a smooth function of two variables, (x,y,f(x,y)) dropping the z axis is a chart. Not sure if f is only continuous.

 

[saminator910]

Not sure if f is only continuous.

What do you mean here?

 

[lavinia]

What do you mean here?

The projection onto the xy-plane needs to be a homeomorphism. If f were discontinuous e.g. it had a huge jump in it so the graph was broken into two pieces, then this would not be a homeomorphism. If f is merely continuous rather than differentiable or smooth I am not sure. Probably.

What if f were continuous but also wild such as the norm of a two dimensional Brownian motion or the z coordinate of a space filling function. I guess the projection would be a homeomorphism.

 

[saminator910]

When is it trivial to prove a homeomorphism?

Also, referring to jgen's first example, while defining charts for the 2-sphere to drop the z coordinate, why does one need 8 charts? It seems that you could do just the same with 2 charts, and there overlap would be the circle that sits on ℝ². The first would be the positive part of the sphere and zero, the second would be the negative and zero. ie.

U={(x1,x2,x3)\niS²|x3≥0}

V={(x1,x2,x3)\niS²|x3≤0}

You could even use the same function to map these sets to ℝ²

 

[jgens]

Also, referring to jgen's first example, while defining charts for the 2-sphere to drop the z coordinate, why does one need 8 charts?

The example that I gave has 6 charts for the 2-sphere. You can do much better with the stereographic projections (where you only need 2 charts) but those do not descend to charts on the real projective space.

It seems that you could do just the same with 2 charts, and there overlap would be the circle that sits on ℝ². The first would be the positive part of the sphere and zero, the second would be the negative and zero.

You would need these hemispheres to be open and they are not if you extend them as you have.

 

[lavinia]

When is it trivial to prove a homeomorphism?

Also, referring to jgen's first example, while defining charts for the 2-sphere to drop the z coordinate, why does one need 8 charts? It seems that you could do just the same with 2 charts, and there overlap would be the circle that sits on ℝ². The first would be the positive part of the sphere and zero, the second would be the negative and zero. ie.

U={(x1,x2,x3)\niS²|x3≥0}

V={(x1,x2,x3)\niS²|x3≤0}

You could even use the same function to map these sets to ℝ²

If you use only two charts, their intersection is a cylinder which is not homeomorphic to an open subset of the plane.

If f(x,y,) is smooth then the equation z = f(x,y) defines a smooth manifold. One could proof this with the Implicit Function Theorem. Projection onto the xy-plane is differentiable and 1-1.

 

[saminator910]

Yes, I realized those sets were closed shortly after posting, I made some stereographic charts for the 2-sphere, do they look right?

ψ(x₁,x₂,x₃) = (\frac{-x_{1}}{x_{3}-1},\frac{-x_{2}}{x_{3}-1},0)
V=S²-{(0,0,1)}
ψ:V→ℝ²

φ(x₁,x₂,x₃) = (\frac{x_{1}}{x_{3}+1},\frac{x_{2}}{x_{3}+1},0)
U=S²-{(0,0,-1)}
φ:U→ℝ²

Also, say I wanted to prove that this is a differential structure on the sphere, so It needs to have the coordinate transformation

ψ φ^-1: φ(U\capV) →R2

But how would I find φ^-1??