Vipul at 21:14, 12 September 2008

2008-09-12T21:14:37Z

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 ==Definition==
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 * It is the one-point compactification of <math>\mathbb{C}</math>, the one additional point being a ''point at infinity'', denoted <math>\infty</math>. With respect to the stereographic projection, the point at infinity is identified with the north pole.
 * It is the [[complex projective line]]: the set of complex lines through the origin in <math>\mathbb{C}^2</math>.

Vipul: 1 revision

2008-05-18T19:18:12Z

1 revision

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Vipul: New page: ==Definition== ===As a Riemann surface=== The Riemann sphere is the topological space $S^2 \subset \R^3$ (the unit sphere in 3-space) with the following atlas of charts: * Th...

2008-04-20T20:11:29Z

New page: ==Definition== ===As a Riemann surface=== The Riemann sphere is the topological space <math>S^2 \subset \R^3</math> (the unit sphere in 3-space) with the following atlas of charts: * Th...

New page

==Definition==

===As a Riemann surface===

The Riemann sphere is the topological space <math>S^2 \subset \R^3</math> (the unit sphere in 3-space) with the following atlas of charts:

* The stereographic projection that maps the complement of the north pole to <math>\mathbb{C}</math>
* The stereographic projection that maps the complement of the south pole to <math>\mathbb{C}</math>, composed with a reflection about the <math>x</math>-axis in <math>\mathbb{C}</math>

The transition map between these two charts is given by:

<math>z \mapsto \frac{1}{z}</math>

===Other descriptions===

The Riemann sphere is viewed in many of the following ways:

* It is the one-point compactification of <math>\mathbb{C}</math>, the one additional point being a ''point at infinity'', denoted <math>\infty</math>. With respect to the stereographic projection, the point at infinity is identified with the north pole.
* It is the [[complex projective line]]: the set of complex lines through the origin in <math>\mathbb{C}^2</math>.

Riemann sphere - Revision history

Vipul at 21:14, 12 September 2008

Vipul: 1 revision

Vipul: New page: ==Definition== ===As a Riemann surface=== The Riemann sphere is the topological space S^2 \subset \R^3 (the unit sphere in 3-space) with the following atlas of charts: * Th...

Vipul: New page: ==Definition== ===As a Riemann surface=== The Riemann sphere is the topological space $S^2 \subset \R^3$ (the unit sphere in 3-space) with the following atlas of charts: * Th...