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.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Riemann sphere canonical | 13 |
| CP^1 | 2 |
| \hat{\mathbb{C}} | 1 |
| complex projective line | 1 |
| complex projective line ℂℙ¹ | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2364486 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Riemann sphere Context triple: [Riemann surface, example, Riemann sphere]
-
A.
Riemann mapping theorem
The Riemann mapping theorem is a fundamental result in complex analysis stating that any non-empty simply connected open subset of the complex plane (other than the whole plane) can be conformally mapped onto the open unit disk.
-
B.
Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
-
C.
Riemann surfaces
Riemann surfaces are one-dimensional complex manifolds that provide the natural geometric setting for studying complex analytic functions and their multi-valued behavior.
-
D.
Conway sphere
The Conway sphere is a mathematical construct in knot theory used to decompose knots and links into simpler tangles, named after mathematician John Horton Conway.
-
E.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Riemann sphere Target entity description: 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.
-
A.
Riemann mapping theorem
The Riemann mapping theorem is a fundamental result in complex analysis stating that any non-empty simply connected open subset of the complex plane (other than the whole plane) can be conformally mapped onto the open unit disk.
-
B.
Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
-
C.
Riemann surfaces
Riemann surfaces are one-dimensional complex manifolds that provide the natural geometric setting for studying complex analytic functions and their multi-valued behavior.
-
D.
Conway sphere
The Conway sphere is a mathematical construct in knot theory used to decompose knots and links into simpler tangles, named after mathematician John Horton Conway.
-
E.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
- F. None of above. chosen
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 |
complex projective line
ⓘ
linked to:
Riemann sphere
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 |
\hat{\mathbb{C}}
ⓘ
linked to:
Riemann sphere
\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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Riemann sphere Description of subject: 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.
Referenced by (18)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Riemann surfaces
linked to: Riemann sphere
linked to: Riemann sphere
linked to: Riemann sphere
linked to: Riemann sphere
subject linked to:
Blaschke products
subject linked to:
Möbius transformations
subject linked to:
PSL(2,\mathbb{C})
subject linked to:
PSL(2,\mathbb{C})
subject linked to:
PSL(2,\mathbb{C})
linked to: Riemann sphere
subject linked to:
PSL(2,\mathbb{C})
subject linked to:
PSL(2,\mathbb{C})
subject linked to:
Fatou sets