Riemann sphere

E259767

Riemann surface compact Riemann surface complex manifold complex projective line extended complex plane mathematical object one-dimensional complex manifold simply connected surface

The Riemann sphere is the complex plane plus a point at infinity, forming a one-dimensional complex manifold topologically equivalent to a sphere and used to study meromorphic functions and complex analysis.

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All labels observed (3)

Label	Occurrences
Riemann sphere canonical	4
\hat{\mathbb{C}}	1
complex projective line	1

Statements (51)

Predicate	Object
instanceOf	Riemann surface ⓘ compact Riemann surface ⓘ complex manifold ⓘ complex projective line ⓘ extended complex plane ⓘ mathematical object ⓘ one-dimensional complex manifold ⓘ simply connected surface ⓘ
alsoKnownAs	Riemann sphere ⓘ surface form: complex projective line extended complex plane ⓘ projective line over the complex numbers ⓘ
automorphismGroup	Möbius transformations ⓘ fractional linear transformations ⓘ
automorphismGroupIsomorphicTo	PSL(2,\mathbb{C}) ⓘ
constructedBy	stereographic projection of complex plane onto sphere ⓘ
contains	complex plane ⓘ
coordinateModel	complex projective coordinates [z:1] and [1:0] ⓘ
curvature	constant positive curvature in standard metric ⓘ
definedAs	complex plane plus a point at infinity ⓘ
dimension	1 complex dimension ⓘ
distinguishedPoint	infinity ⓘ
EulerCharacteristic	2 ⓘ
fundamentalGroup	trivial group ⓘ
genus	0 ⓘ
hasChart	stereographic projection from north pole ⓘ stereographic projection from south pole ⓘ
hasPoint	point at infinity ⓘ
homotopyType	2-sphere S^2 ⓘ
isCompact	true ⓘ
isConnected	true ⓘ
isOnePointCompactificationOf	complex plane ⓘ
isSimplyConnected	true ⓘ
namedAfter	Bernhard Riemann ⓘ
property	every holomorphic function on it is constant ⓘ every meromorphic function on complex plane extends to holomorphic map to Riemann sphere ⓘ every rational function defines a holomorphic self-map ⓘ
realDimension	2 ⓘ
roleIn	classification of compact Riemann surfaces ⓘ model for one-point compactification of complex plane ⓘ
symbol	Riemann sphere self-linksurface differs ⓘ surface form: \hat{\mathbb{C}} \mathbb{C} \cup \{\infty\} ⓘ
topologicallyEquivalentTo	2-sphere ⓘ unit sphere in \mathbb{R}^3 ⓘ
usedIn	algebraic geometry ⓘ complex analysis ⓘ complex dynamics ⓘ conformal mapping theory ⓘ dynamical systems ⓘ geometric function theory ⓘ projective geometry ⓘ theory of meromorphic functions ⓘ

Referenced by (6)

Full triples — surface form annotated when it differs from this entity's canonical label.

Riemann surfaces → example → Riemann sphere ⓘ

subject surface form: Riemann surface

Kleinian group → actsOn → Riemann sphere ⓘ

Riemann sphere → alsoKnownAs → Riemann sphere ⓘ

this entity surface form: complex projective line

Riemann sphere → symbol → Riemann sphere self-linksurface differs ⓘ

this entity surface form: \hat{\mathbb{C}}

uniformization theorem → classifiesAs → Riemann sphere ⓘ

Picard theorem → relatedTo → Riemann sphere ⓘ