Riemann surfaces

E47348

analytic space complex manifold mathematical concept one-dimensional complex manifold topological space

Riemann surfaces are one-dimensional complex manifolds that provide the natural geometric setting for studying complex analytic functions and their multi-valued behavior.

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All labels observed (5)

Label	Occurrences
Riemann surfaces canonical	8
Riemann surface	3
Riemann surface theory	2
Die Idee der Riemannschen Fläche	1
Riemann surfaces and analytic relations	1

Statements (58)

Predicate	Object
instanceOf	analytic space ⓘ complex manifold ⓘ mathematical concept ⓘ one-dimensional complex manifold ⓘ topological space ⓘ
admits	conformal metric ⓘ holomorphic 1-forms ⓘ holomorphic functions ⓘ meromorphic functions ⓘ
classifiedBy	genus in the compact case ⓘ
compactCaseCorrespondsTo	smooth projective algebraic curves over the complex numbers ⓘ
definedOver	complex numbers ⓘ
dimension	one complex dimension ⓘ
equivalentTo	one-dimensional complex analytic manifold ⓘ one-dimensional complex manifold with holomorphic transition maps ⓘ
example	Riemann sphere ⓘ compact Riemann surface of genus g ⓘ complex plane ⓘ complex torus ⓘ unit disk with its complex structure ⓘ
field	algebraic geometry ⓘ complex analysis ⓘ differential geometry ⓘ topology ⓘ
generalizes	branched coverings of the complex plane ⓘ open subsets of the complex plane ⓘ
hasInvariant	Euler’s polyhedron formula ⓘ surface form: Euler characteristic complex structure ⓘ conformal structure ⓘ fundamental group ⓘ genus ⓘ
hasProperty	Hausdorff ⓘ connected (usually assumed in the definition) ⓘ orientable ⓘ second countable ⓘ
hasTheorem	Hodge decomposition for compact Riemann surfaces ⓘ Riemann–Hurwitz formula ⓘ Riemann–Roch theorem ⓘ uniformization theorem ⓘ
localModel	open subsets of the complex plane ⓘ
namedAfter	Bernhard Riemann ⓘ
orientationInducedBy	complex structure ⓘ
realDimension	two real dimensions ⓘ
relatedTo	Fuchsian group ⓘ Kleinian group ⓘ algebraic curve ⓘ complex curve ⓘ moduli space of curves ⓘ
structure	atlas of charts to the complex plane ⓘ
transitionMaps	holomorphic functions ⓘ
usedFor	Teichmüller theory ⓘ analytic continuation ⓘ moduli problems in algebraic geometry ⓘ monodromy theory ⓘ studying complex analytic functions ⓘ studying multi-valued analytic functions ⓘ theory of algebraic curves ⓘ uniformization theory ⓘ

Referenced by (15)

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

Bernhard Riemann → knownFor → Riemann surfaces ⓘ

Hermann Weyl → notableWork → Riemann surfaces ⓘ

this entity surface form: Die Idee der Riemannschen Fläche

Georg → notableWork → Riemann surfaces ⓘ

subject surface form: Georg Friedrich Bernhard Riemann

Friedrich → notableWork → Riemann surfaces ⓘ

subject surface form: Friedrich Bernhard Riemann

Carathéodory metric → definedOn → Riemann surfaces ⓘ

this entity surface form: Riemann surface

Lars Ahlfors → fieldOfWork → Riemann surfaces ⓘ

modular group PSL(2,Z) → relatedTo → Riemann surfaces ⓘ

subject surface form: PSL(2,ℤ)

Jacobi operator → relatedTo → Riemann surfaces ⓘ

Hilbert’s twenty-second problem → hasSubjectHeading → Riemann surfaces ⓘ

this entity surface form: Riemann surfaces and analytic relations

Seiberg–Witten theory → involves → Riemann surfaces ⓘ

Teichmüller theory → fieldOfStudy → Riemann surfaces ⓘ

Kleinian group → associatedWith → Riemann surfaces ⓘ

this entity surface form: Riemann surface

uniformization theorem → usesConcept → Riemann surfaces ⓘ

this entity surface form: Riemann surface

Brill–Noether theory → relatedTo → Riemann surfaces ⓘ

this entity surface form: Riemann surface theory

Picard theorem → hasProofMethod → Riemann surfaces ⓘ

this entity surface form: Riemann surface theory