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How To define charts and atlases 
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#1
Dec3112, 09:34 PM

P: 94

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 mmanifold 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 1manifold? 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.



#2
Jan113, 04:17 AM

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P: 26,148

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 


#3
Jan113, 09:34 AM

P: 280

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



#4
Jan113, 11:08 AM

P: 94

How To define charts and atlases
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?



#5
Jan113, 02:38 PM

PF Gold
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#6
Jan113, 07:49 PM

P: 94

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 nSphere from ℝn+1 to ℝn by leaving out a coordinate, correct? So on a 2sphere 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?



#7
Jan113, 08:38 PM

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#8
Jan113, 09:28 PM

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


#9
Jan113, 09:39 PM

P: 94

I guess I'm just confused as to how the mapping should "look" afterward, for example, since one need only prove that a mmanifold is homeomorphic to ℝ^{m}, so for the hyperbolic paraboloid,
h={(x_{1},x_{2},x_{3})x_{1}^{2}x_{2}^{2}x_{3}=0} one chart should be able to map that onto ℝ^{2}, correct? so could you use that same method of removing the x_{3} coordinate? that would literally produce ℝ^{2}, but it seems too trivial. but even the fact that the hyperbolic paraboloid is composed of a continuous function from ℝ^{2} to ℝ means that it is a homeomorphism of ℝ^{2}, correct? 


#10
Jan113, 09:59 PM

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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. 


#11
Jan113, 10:03 PM

P: 94




#12
Jan113, 10:06 PM

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P: 1,724

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. 


#13
Jan113, 10:19 PM

P: 94

When is it trivial to prove a homeomorphism?
Also, referring to jgen's first example, while defining charts for the 2sphere 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 ℝ^{2}. 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)[itex]\ni[/itex]S^{2}x3≥0} V={(x1,x2,x3)[itex]\ni[/itex]S^{2}x3≤0} You could even use the same function to map these sets to ℝ^{2} 


#14
Jan113, 10:28 PM

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#15
Jan113, 10:30 PM

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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 xyplane is differentiable and 11. 


#16
Jan213, 06:47 PM

P: 94

Yes, I realized those sets were closed shortly after posting, I made some stereographic charts for the 2sphere, do they look right?
ψ(x_{1},x_{2},x_{3}) = ([itex]\frac{x_{1}}{x_{3}1}[/itex],[itex]\frac{x_{2}}{x_{3}1}[/itex],0) V=S^{2}{(0,0,1)} ψ:V→ℝ^{2} φ(x_{1},x_{2},x_{3}) = ([itex]\frac{x_{1}}{x_{3}+1}[/itex],[itex]\frac{x_{2}}{x_{3}+1}[/itex],0) U=S^{2}{(0,0,1)} φ:U→ℝ^{2} 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[itex]\cap[/itex]V) →R2 But how would I find φ^{1}?? 


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