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.

Aliases (2)

Riemann surface ×58
Die Idee der Riemannschen Fläche ×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 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 (4)

Subject (surface form when different)	Predicate
Friedrich Bernhard Riemann → Georg Friedrich Bernhard Riemann → Hermann Weyl ("Die Idee der Riemannschen Fläche") →	notableWork
Bernhard Riemann →	knownFor