Homogeneous Riemann surface - Revision history

Vipul at 21:00, 12 September 2008

2008-09-12T21:00:01Z

← Older revision		Revision as of 21:00, 12 September 2008
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	* The [[conformal automorphism group]] acts transitively on the points of the Riemann surface		* The [[conformal automorphism group]] acts transitively on the points of the Riemann surface
	* Given any two points of the Riemann surface, there is a bijective biholomorphic mapping from the surface to itself that sends the first point to the second.		* Given any two points of the Riemann surface, there is a bijective biholomorphic mapping from the surface to itself that sends the first point to the second.

			==Facts==

			* A [[compact Riemann surface]] is homogeneous if and only if it has genus zero or one. This follows from the fact that compact Riemann surfaces of higher genus have finite automorphism groups (alternatively, it follows from the fact that higher genus Riemann surfaces have certain special points called [[Weierstrass point]]s). {{further\|[[Compact and homogeneous iff genus zero or one]]}}
			* Any simply connected Riemann surface is homogeneous. This follows from the classification of simply connected Riemann surfaces by the uniformization theorem: the only simply connected Riemann surfaces are the [[open unit disk]], the [[Riemann sphere]], and the [[complex plane]].

Vipul: 1 revision

2008-05-18T19:13:46Z

1 revision

← Older revision	Revision as of 19:13, 18 May 2008
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Vipul: New page: {{Riemann surface property}} ==Definition== A Riemann surface is termed '''homogeneous''', '''transitive''', '''conformally homogeneous''', or '''conformally transitive''' if it sati...

2008-04-20T23:25:42Z

New page: {{Riemann surface property}} ==Definition== A Riemann surface is termed '''homogeneous''', '''transitive''', '''conformally homogeneous''', or '''conformally transitive''' if it sati...

New page

{{Riemann surface property}}

==Definition==

A [[Riemann surface]] is termed '''homogeneous''', '''transitive''', '''conformally homogeneous''', or '''conformally transitive''' if it satisfies the following equivalent conditions:

* The [[conformal automorphism group]] acts transitively on the points of the Riemann surface
* Given any two points of the Riemann surface, there is a bijective biholomorphic mapping from the surface to itself that sends the first point to the second.