Charts of a torus (and other manifolds)

[mcafej]

Ok, so this relates to my homework, but I really can't find an answer anywhere, so this is more of a general question. First off, what does a "chart" of a manifold look like? Is it a set, a function, a drawing, a table, what?! I have found so many things about charts, but nothing shows what they actually look like (either that or I am just not recognizing them). Secondly, using stereographical projection, what are the 4 charts of a torus? I saw something that looked something like

x¹ = ((cosΘ, sinΘ), (cosφ, sinφ))= Θ
x² = ((cosΘ, sinΘ), (cosφ, sinφ))= φ

and the other charts are similar. Would the other charts just be

y¹ = ((cosΘ, sinΘ), (cosφ, sinφ))= Θ
y² = ((cosΘ, sinΘ), (cosφ, sinφ))= φ

I'm just really confused on this topic and I would like to gain a little more confidence before going into my final. Are the "charts" that I gave actually charts, and are they the ones that describe a torus?

 

[lavinia]

On the sphere of radius 1 in 3 space centered at the origin you have four charts. Project the upper hemisphere onto the xy-plane by dropping the z coordinate. Project the lower hemisphere the same way. then rotate the sphere 90 degrees and do it again.

 

[Vargo]

A chart is a map that you use to navigate your way through some area. For example, the USGS makes a ton of maps so that you can look at the appropriate chart to check out the local terrain/landmarks. Usually, charts are marked to show latitude and longitude. These are coordinates that tell you where you are on that chart.
An atlas is a book of charts that covers all the places of interest.

The meaning of chart/atlas in manifold theory is the same in spirit. A chart is a map of the area near a point and the points on the charts have rectangular coordinates that tell you where you are on the chart. Given a manifold there are an infiinte number of charts. Just like on the globe there are an infinite number of possible charts because a chart is determined by the exact area I want to map and also by the manner in which I want to assign coordinates to the points on the chart (You dont' have to use latitude/longitude, you could make up another way to assign coordinates).

Formally speaking, a chart consists of an open set U in the manifold, and a bijective mapping
$\varphi: U\rightarrow \mathbb{R}^n .$
The image of the mapping is your flat "map" of the region U.

The mapping you described does give a chart of the torus though to be precise you need to specify the domain and range so that you have a 1-1 mapping. The fact that you need more than 1 chart is a reflection of the fact that a single flat map cannot smoothly cover the torus unless you wrap it on itself.

 

[lavinia]

If you think of the torus as a rectangle in the plane with opposite edges identified then you can get a coordinate system as

1) the interior of the rectangle
2) an open square around the corner,
3) an open rectangular strip along two perpendicular edges excluding the end points

Each of these is a diffeomorphism of an open neighborhood on the torus into an open neighborhood in the plane

 

[blue_raver22]

A chart of a manifold is a mathematical tool used to map points on the manifold to points in a coordinate system. It is typically represented as a function that takes a point on the manifold as input and outputs its coordinates in the coordinate system. In other words, a chart is a way to represent the points on a manifold in a more familiar coordinate system.

In the case of a torus, a chart would be a function that maps points on the surface of the torus to points in a 2D coordinate system, such as the x-y plane. The charts you have provided are examples of such functions, but they are not the only ones that can be used to describe a torus. The stereographic projection method you mentioned is one way to create charts for a torus, but there are other methods as well.

The four charts you have provided are indeed charts for a torus, but they are not the only ones. You can create an infinite number of charts for a torus, as long as they cover the entire surface of the torus without any overlaps or gaps.

I would recommend consulting with your professor or looking for additional resources to gain a better understanding of charts and their role in describing manifolds. It is important to have a solid understanding of this concept in order to successfully complete your homework and final exam.