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wrobel
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the sphere is a compact set and a chart is notcianfa72 said:
the sphere is a compact set and a chart is notcianfa72 said:Summary:: Formal proof that it does not exist a global coordinate chart on a 2-sphere
So, from a formal mathematical point of view, how to prove it ? Just because there is not a (global) homeomorphism between the 2-sphere and the Euclidean plane ? Thanks.
Just to add some detail to your claim. Suppose there is a single chart for the 2-sphere: we can use such chart to define a (one-chart) differentiable atlas for the 2-sphere (by definition a chart is compatible with itself).jbergman said:If there is a single chart for the 2-sphere then we could use that chart to define a non-vanishing vector field for every point on the sphere, i.e., just a vector in the direction of one of the coordinate axes.
We know this is impossible by the hairy ball theorem.
I haven't read the details but by Borsuk Up an, it is not possible.cianfa72 said:Let's try to visualize it in 3D space. Suppose the 2-sphere is placed in 3D such that the plane ##x=0## is tangent to it on the "left side". Then, starting from the left, for each half-circle on ##x=c , c>0## planes assign coordinate ##s=c## to one half-circle and ##s=-c## to the other (proceed this way up to the 2-sphere "right side").
As above I believe it should result in a one-to-one map, however as you pointed out the image on ##\mathbb R^2## should be not an open set in ##\mathbb R^2## standard topology.
The goal was try to build a one-to-one map for the 2-sphere. As we know, however, it can never be a (global) chart for it.
Make sense ? Thank you in advance.
A global coordinate chart on a 2-sphere is a way to represent points on a 2-sphere using two coordinates. It is similar to a map projection, where a curved surface is flattened onto a 2D plane. The coordinates are typically latitude and longitude, but other systems can also be used.
A global coordinate chart on a 2-sphere is useful for navigation and location-based calculations on a spherical surface. It allows for easy visualization and measurement of distances and angles between points on the 2-sphere.
One limitation is that it can only represent points on a 2-sphere, and not on other curved surfaces. Another limitation is that it can introduce distortions and inaccuracies, especially near the poles of the 2-sphere.
A global coordinate chart on a 2-sphere is similar to the coordinate system used to map locations on the Earth's surface. However, the Earth is not a perfect sphere, so more complex coordinate systems, such as the WGS84 system, are used to account for its irregular shape.
Yes, a global coordinate chart on a 2-sphere can be used for any size of 2-sphere, as long as the coordinates are properly scaled. This means that the same coordinate system can be used for a small object, like a marble, as well as for a large object, like the Earth.