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advanced_tools:manifold [2018/03/06 15:47]
jakobadmin [Examples]
advanced_tools:manifold [2020/01/11 03:46] (current)
sebsonf Radius equation
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 <tabbox Why is it interesting?> ​ <tabbox Why is it interesting?> ​
 +Manifolds are especially important in [[models:​general_relativity|General Relativity]]. Through massive objects, spacetime is curved and no longer flat. Manifolds are the correct mathematical tool to describe spacetime when it isn't flat. 
  
 <tabbox Layman> ​ <tabbox Layman> ​
  
-<note tip> +  * http://​bastian.rieck.me/​blog/​posts/​2019/​manifold/
-Explanations in this section should contain no formulas, but instead, colloquial things like you would hear them during a coffee break or at a cocktail party. +
-</note>+
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 <tabbox Student> ​ <tabbox Student> ​
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 +--> The Two-Sphere#
  
 +{{ :​advanced_tools:​spheremanifoldexample2.png?​nolink&​400|}}
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 +Another example of a manifold is the two-sphere. The two-sphere $S^2$ is defined as the set of points in $R^3$ for which $x^2+y^2+z^2=const$ holds,​ where $const$ is, of course, the radius of the sphere. The two-sphere is two-dimensional because the definition involves $3$ coordinates and one condition, which eliminates one degree of freedom. Therefore to see that the sphere is a manifold we need a map onto $R^2$. This map is given by the usual spherical coordinates.
 +Almost all points on the surface of the sphere can be identified unambiguously with a coordinate combination of the form $(\varphi, \theta)$. Almost all! Where is the pole $\varphi=0$ mapped to? There is no one-to-one identification possible because the pole is mapped to a whole line, as indicated in the image.Therefore this map does not work for the complete sphere and we need another map in the neighborhood of the pole to describe things there. A similar problem appears for the map on the semicircle $\theta=0$.  Each point can be mapped to in the $R^2$ to $\theta=0$ and $\theta=2 \pi$, which is again not a one-to-one map. This illustrates the fact that for manifolds there is in general not one coordinate system for all points of the manifold, only local coordinates,​ which are valid in some neighborhood. This is no problem because a manifold is defined to look locally like $R^n$.
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 +The spherical coordinate map is only valid in the open neighborhood $0< \varphi <\pi , 0< \theta < 2 \pi$ and we need a second map to cover the whole sphere. We can use for example a second spherical coordinate system with a different orientation,​ such that the problematic poles lie at different points for this map and no longer at $\varphi=0$. With this second map, every point of the sphere has a map onto $R^2$ and the two-sphere can be seen to be a manifold.
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 +<--
  
 +Another trivial example for a manifold is of course $R^n$.
 <tabbox FAQ> ​ <tabbox FAQ> ​
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advanced_tools/manifold.1520347644.txt.gz · Last modified: 2018/03/06 14:47 (external edit)