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| {{for|the Riemann surface of a subring of a field|Zariski–Riemann space}}
| | You never know exactly what all video game world may contain. There are horrors to bad in every corner and cranny. The post includes advice close to optimizing your gaming year with tricks and indications you might not be particularly aware of. Go forward reading for more detail.<br><br>If you are purchasing a game your child, appear for one that allows several individuals to perform together. Gaming might just be a singular activity. Nonetheless, it's important to guidance your youngster to continually be societal, and multiplayer battle of clans trucos events can do that. They allow siblings and also buddies to all seated and laugh and play together.<br><br>Small business inside your games when you find yourself succesfully done playing them. A variety of retailers provide discount monthly premiums or credit score to help your next buy when ever you business your clash of clans sur pc tlcharger about. If you have any sort of questions relating to where and ways to make use of how to hack clash of clans ([http://prometeu.net prometeu.net]), you could contact us at our webpage. You can receive the next online video game you would like intended for the affordable price shortly after you try this. All things considered, they don't need the video media games as soon since you defeat them.<br><br>In cases where you're playing a game online, and you range across another player of which seems to be aggravating other players (or you, in particular) intentionally, never will take it personally. This is called "Griefing," and it's the spot the [http://photo.net/gallery/tag-search/search?query_string=equivalent equivalent] of Internet trolling. Griefers are mearly out for negative attention, and you give people what they're looking to obtain if you interact together. Don't get emotionally utilized in what's happening in addition to simply try to ignore it.<br><br>Sensei Wars, the feudal Japan-themed Clash of Clans Power tips attacker from 2K, has aloof accustomed its aboriginal agreeable amend again it is really barrage on iOS aftermost 12 ,.<br><br>Conserve some money on your main games, think about opting-in into a assistance in which you can rent payments programs from. The premium of these lease commitments for the year is now normally under the cost of two video gaming applications. You can preserve the match titles until you do more than them and simply dispatch out them back once and purchase another one particular particular.<br><br>Right there is a "start" button to click on all over the wake of getting in the wanted traits. When you start reduced Clash of Clans chop hack cheats tool, hang on around for a 50 % of moment, slammed refresh and you will also have the means everyone needed. There must be nothing at all result in in working with thjis hack and cheats method. Make utilization of all the Means that someone have, and exploit the idea 2013 Clash of Clans hack obtain! Explanation why fork out for difficult or gems when a person will can get the awaited things with this piece of equipment! Sprint and get your proprietary Clash related to Clans hack software today. The required items are only a a handful of of clicks absent. |
| [[Image:Riemann sqrt.jpg|thumb|right| Riemann surface for the function ''ƒ''(''z'') = √''z''. The two horizontal axes represent the real and imaginary parts of ''z'', while the vertical axis represents the real part of √''z''. For the imaginary part of √''z'', rotate the plot 180° around the vertical axis.]]
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| In [[mathematics]], particularly in [[complex analysis]], a '''Riemann surface''', first studied by and named after [[Bernhard Riemann]], is a one-dimensional [[complex manifold]]. Riemann surfaces can be thought of as "deformed versions" of the [[complex plane]]: locally near every point they look like patches of the complex plane, but the global [[topology]] can be quite different. For example, they can look like a [[sphere]] or a [[torus]] or a couple of sheets glued together.
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| The main point of Riemann surfaces is that [[holomorphic function]]s may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially [[multi-valued function]]s such as the [[square root]] and other [[algebraic function]]s, or the [[natural logarithm|logarithm]].
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| Every Riemann surface is a two-dimensional real analytic [[manifold]] (i.e., a [[surface]]), but it contains more structure (specifically a [[Complex Manifold|complex structure]]) which is needed for the unambiguous definition of holomorphic functions. A two-dimensional real manifold can be turned into a Riemann surface (usually in several inequivalent ways) if and only if it is [[orientable]] and metrizable. So the sphere and torus admit complex structures, but the [[Möbius strip]], [[Klein bottle]] and [[projective plane]] do not.
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| Geometrical facts about Riemann surfaces are as "nice" as possible, and they often provide the intuition and motivation for generalizations to other curves, manifolds or varieties. The [[Riemann–Roch theorem]] is a prime example of this influence.
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| == Definitions ==
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| There are several equivalent definitions of a Riemann surface.
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| # A Riemann surface ''X'' is a [[complex manifold]] of complex [[dimension]] one. This means that ''X'' is a [[Hausdorff space|Hausdorff]] [[topological space]] endowed with an [[atlas (topology)|atlas]]: for every point ''x'' ∈ ''X'' there is a [[open set|neighbourhood]] containing ''x'' [[homeomorphic]] to the unit disk of the [[complex plane]]. The map carrying the structure of the complex plane to the Riemann surface is called a ''chart''. Additionally, the [[transition map]]s between two overlapping charts are required to be [[Holomorphic function|holomorphic]].
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| # A Riemann surface is an [[oriented manifold|oriented]] [[manifold]] of (real) dimension two – a two-sided [[surface]] – together with a [[conformal structure]]. Again, manifold means that locally at any point ''x'' of ''X'', the space is supposed to be like the real plane. The supplement "Riemann" signifies that ''X'' is endowed with an additional structure which allows [[angle]] measurement on the manifold, namely an [[equivalence class]] of so-called [[Riemannian metric]]s. Two such metrics are considered [[conformally equivalent|equivalent]] if the angles they measure are the same. Choosing an equivalence class of metrics on ''X'' is the additional datum of the conformal structure.
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| A complex structure gives rise to a conformal structure by choosing the standard [[Euclidean metric]] given on the complex plane and transporting it to ''X'' by means of the charts. Showing that a conformal structure determines a complex structure is more difficult.<ref>See {{Harvard citations|author=Jost|year=2006|loc=Ch. 3.11}} for the construction of a corresponding complex structure.</ref>
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| {{further2|[[complex manifold]] and [[conformal geometry]]}}
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| == Examples ==
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| [[Image:Riemann sphere1.jpg|thumb|left|150px|The Riemann sphere.]]
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| *The [[complex plane]] '''C''' is the most basic Riemann surface. The map ''f''(''z'') = ''z'' (the identity map) defines a chart for '''C''', and {''f''} is an [[atlas (topology)|atlas]] for '''C'''. The map ''g''(''z'') = ''z<sup>*</sup>'' (the [[complex conjugate|conjugate]] map) also defines a chart on '''C''' and {''g''} is an atlas for '''C'''. The charts ''f'' and ''g'' are not compatible, so this endows '''C''' with two distinct Riemann surface structures. In fact, given a Riemann surface ''X'' and its atlas ''A'', the conjugate atlas ''B'' = {''f<sup>*</sup>'' : ''f'' ∈ ''A''} is never compatible with ''A'', and endows ''X'' with a distinct, incompatible Riemann structure.
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| *In an analogous fashion, every [[open subset]] of the complex plane can be viewed as a Riemann surface in a natural way. More generally, every open subset of a Riemann surface is a Riemann surface.
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| *Let ''S'' = '''C''' ∪ {∞} and let ''f''(''z'') = ''z'' where ''z'' is in ''S'' \ {∞} and ''g''(''z'') = 1 / ''z'' where ''z'' is in ''S'' \ {0} and 1/∞ is defined to be 0. Then ''f'' and ''g'' are charts, they are compatible, and { ''f'', ''g'' } is an atlas for ''S'', making ''S'' into a Riemann surface. This particular surface is called the '''[[Riemann sphere]]''' because it can be interpreted as wrapping the complex plane around the sphere. Unlike the complex plane, it is [[compact space|compact]].
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| [[Image:Torus.png|right|thumb|150px|A torus.]]
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| *The theory of compact Riemann surfaces can be shown to be equivalent to that of projective [[algebraic curve]]s that are defined over the complex numbers and non-singular. For example, the [[torus]] '''C'''/('''Z''' + '''τ Z'''), where '''τ''' is a complex non-real number, corresponds, via the [[Weierstrass elliptic function]] associated to the [[lattice (group)|lattice]] '''Z''' + '''τ''' '''Z''', to an [[elliptic curve]] given by an equation
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| ::''y''<sup>2</sup> = ''x''<sup>3</sup> + ''a x'' + ''b''.
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| :Tori are the only Riemann surfaces of [[genus (mathematics)|genus]] one, surfaces of higher genera ''g'' are provided by the [[hyperelliptic surface]]s
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| ::''y''<sup>2</sup> = ''P''(''x''),
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| :where ''P'' is a complex [[polynomial]] of degree 2''g'' + 1.
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| *Important examples of non-compact Riemann surfaces are provided by [[analytic continuation]].
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| <gallery>
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| File:Riemann surface arcsin.jpg|''f''(''z'') = arcsin ''z''
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| File:Riemann surface log.jpg|''f''(''z'') = log ''z''
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| File:Riemann surface sqrt.jpg|''f''(''z'') = ''z''<sup>1/2</sup>
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| File:Riemann surface cube root.jpg|''f''(''z'') = ''z''<sup>1/3</sup>
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| File:Riemann surface 4th root.jpg|''f''(''z'') = ''z''<sup>1/4</sup>
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| </gallery>
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| == Further definitions and properties ==
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| As with any map between complex manifolds, a [[function (mathematics)|function]] ''f'': ''M'' → ''N'' between two Riemann surfaces ''M'' and ''N'' is called ''[[holomorphic]]'' if for every chart ''g'' in the [[atlas (topology)|atlas]] of ''M'' and every chart ''h'' in the atlas of ''N'', the map ''h'' o ''f'' o ''g''<sup>−1</sup> is holomorphic (as a function from '''C''' to '''C''') wherever it is defined. The composition of two holomorphic maps is holomorphic. The two Riemann surfaces ''M'' and ''N'' are called ''biholomorphic'' (or ''conformally equivalent'' to emphasize the conformal point of view) if there exists a [[bijective]] holomorphic function from ''M'' to ''N'' whose inverse is also holomorphic (it turns out that the latter condition is automatic and can therefore be omitted). Two conformally equivalent Riemann surfaces are for all practical purposes identical.
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| ===Orientability===
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| We noted in the preamble that all Riemann surfaces, like all complex manifolds, are [[orientable]] as a real manifold. The reason is that for complex charts ''f'' and ''g'' with transition function ''h'' = ''f''(''g''<sup>−1</sup>(''z'')) we can consider ''h'' as a map from an open set of '''R'''<sup>2</sup> to '''R'''<sup>2</sup> whose [[Jacobian]] in a point ''z'' is just the real linear map given by multiplication by the complex number ''h''<nowiki>'</nowiki>(''z''). However, the real [[determinant]] of multiplication by a complex number ''α'' equals |''α''|<sup>2</sup>, so the Jacobian of ''h'' has positive determinant. Consequently the complex atlas is an oriented atlas.
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| === Functions ===
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| Every non-compact Riemann surface admits non-constant holomorphic functions (with values in '''C'''). In fact, every non-compact Riemann surface is a [[Stein manifold]].
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| In contrast, on a compact Riemann surface ''X'' every holomorphic function with value in '''C''' is constant due to the [[maximum principle]]. However, there always exists non-constant [[meromorphic]] functions (holomorphic functions with values in the [[Riemann sphere]] '''C''' ∪ {∞}). More precisely, the [[function field of an algebraic variety|function field]] of ''X'' is a finite [[field extension|extension]] of '''C'''(''t''), the function field in one variable, i.e. any two meromorphic functions are algebraically dependent. This statement generalizes to higher dimensions, see {{harvtxt|Siegel|1955}}.
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| ==Analytic vs. algebraic==
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| The above fact about existence of nonconstant meromorphic functions can be used to show that any compact Riemann surface is a [[projective variety]], i.e. can be given by [[polynomial]] equations inside a [[projective space]]. Actually, it can be shown that every compact Riemann surface can be [[immersion (mathematics)|embedded]] into [[complex projective space|complex projective 3-space]]. This is a surprising theorem: Riemann surfaces are given by locally patching charts. If one global condition, namely compactness, is added, the surface is necessarily algebraic. This feature of Riemann surfaces allows to study them with either the means of [[analytic geometry|analytic]] or [[algebraic geometry]]. The corresponding statement for higher-dimensional objects is false, i.e. there are compact complex 2-manifolds which are not algebraic. On the other hand, every projective complex manifold is necessarily algebraic, see [[Algebraic geometry and analytic geometry#Chow.27s theorem|Chow's theorem]].
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| As an example, consider the torus ''T'' := '''C'''/('''Z''' + ''τ'' '''Z'''). The Weierstrass function <math>\wp_\tau(z)</math> belonging to the lattice '''Z''' + ''τ'' '''Z''' is a [[meromorphic function]] on ''T''. This function and its derivative <math>\wp_\tau'(z)</math> [[Generating set|generate]] the function field of ''T''. There is an equation
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| :<math>
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| [\wp'(z)]^2=4[\wp(z)]^3-g_2\wp(z)-g_3,</math>
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| where the coefficients ''g''<sub>2</sub> and ''g''<sub>3</sub> depend on τ, thus giving an elliptic curve ''E''<sub>τ</sub> in the sense of algebraic geometry. Reversing this is accomplished by the [[j-invariant]] ''j''(''E''), which can be used to determine ''τ'' and hence a torus.
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| == Classification of Riemann surfaces ==
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| The realm of Riemann surfaces can be divided into three regimes: hyperbolic, parabolic and elliptic Riemann surfaces, with the distinction given by the [[uniformization theorem]]. Geometrically, these correspond to negative curvature, zero curvature/flat, and positive curvature: stating the uniformization theorem in terms of conformal geometry, every connected Riemann surface ''X'' admits a unique [[completeness (topology)|complete]] 2-dimensional real [[Riemannian manifold|Riemann metric]] with constant [[curvature]] −1, 0 or 1 inducing the same conformal structure – every metric is conformally equivalent to a constant curvature metric. The surface ''X'' is called '''hyperbolic''', '''parabolic''', and '''elliptic''', respectively.
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| For [[simply connected]] Riemann surfaces, the uniformization theorem states that every simply connected Riemann surface is conformally equivalent to one of the following:
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| ;elliptic: the Riemann sphere <math>\widehat{\mathbf{C}} := \mathbf{C} \cup\{\infty\}</math>, also denoted [[complex projective line|'''P'''<sup>1</sup>'''C''']]
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| ;parabolic: the complex plane '''C''', or
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| ;hyperbolic: the [[open disk]] ''D'' := {''z'' ∈ '''C''' : |''z''| < 1} or equivalently the [[upper half-plane]] ''H'' := {''z'' ∈ '''C''' : [[imaginary part|Im]](''z'') > 0}.
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| The existence of these three types parallels the several [[non-Euclidean geometry|non-Euclidean geometries]].
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| The general technique of associating to a manifold ''X'' its [[universal cover]] ''Y'', and expressing the original ''X'' as the [[quotient (topology)|quotient]] of ''Y'' by the group of [[deck transformation]]s gives a first overview over Riemann surfaces.
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| === Elliptic Riemann surfaces === | |
| By definition, these are the surfaces ''X'' with constant curvature +1. The [[Riemann sphere]] '''C''' ∪ {∞} is the only example. ([[Elliptic function]]s are examples of parabolic Riemann surfaces. The naming comes from the history: elliptic functions are associated to [[elliptic integrals]], which in turn show up in calculating the [[circumference]] of [[ellipse]]s).
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| === Parabolic Riemann surfaces ===
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| By definition, these are the surfaces ''X'' with constant curvature 0. Equivalently, by the uniformization theorem, the universal cover of ''X'' has to be the complex plane.
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| There are then three possibilities for ''X''. It can be the plane itself, the punctured plane (or cylinder), or a [[torus]]
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| :''T'' := '''C''' / ('''Z''' ⊕ τ'''Z''').
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| The set of representatives of the cosets are called [[fundamental domain]]s.
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| Care must be taken insofar as two tori are always [[homeomorphic]], but in general not biholomorphic to each other. This is the first appearance of the problem of moduli. The modulus of a torus can be captured by a single complex number τ with positive imaginary part. In fact, the marked moduli space ([[Teichmüller space]]) of the torus is biholomorphic to the upper half-plane or equivalently the open unit disk.
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| === Hyperbolic Riemann surfaces ===
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| The Riemann surfaces with curvature −1 are called ''hyperbolic''. This group is the "biggest" one.
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| The celebrated [[Riemann mapping theorem]] states that any simply connected strict subset of the complex plane is biholomorphic to the unit disk. Therefore the open disk with the Poincaré-metric of constant curvature −1 is the local model of any hyperbolic Riemann surface. According to the uniformization theorem above, all hyperbolic surfaces are quotients of the unit disk.
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| Examples include all surfaces with genus ''g'' > 1 such as hyper-elliptic curves.
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| For every hyperbolic Riemann surface, the fundamental group is isomorphic to a [[Fuchsian group]], and thus the surface can be modelled by a [[Fuchsian model]] '''H'''/Γ where '''H''' is the [[upper half-plane]] and Γ is the Fuchsian group. The set of representatives of the cosets of '''H'''/Γ are [[free regular set]]s and can be fashioned into metric [[fundamental polygon]]s. Quotient structures as '''H'''/Γ are generalized to [[Shimura variety|Shimura varieties]].
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| Unlike elliptic and parabolic surfaces, no classification of the hyperbolic surfaces is possible. Any connected open strict subset of the plane gives a hyperbolic surface; consider the plane minus a [[Cantor set]]. A classification ''is'' possible for '''surfaces of finite type''': those isomorphic to a compact surface with a finite number of points removed. Any one of these has a finite number of moduli and so a finite dimensional Teichmüller space. The [[moduli space|problem of moduli]] (solved by [[Lars Ahlfors]] and extended by [[Lipman Bers]]) was to justify Riemann's claim that for a closed surface of genus '''''g''''', '''3''g'' − 3''' complex parameters suffice.
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| When a hyperbolic surface is compact, then the total area of the surface is 4π(''g'' − 1), where ''g'' is the [[genus (mathematics)|genus]] of the surface; the area is obtained by applying the [[Gauss-Bonnet theorem]] to the area of the fundamental polygon.
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| == Maps between Riemann surfaces ==
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| The geometric classification is reflected in maps between Riemann surfaces,
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| as detailed in [[Liouville's theorem (complex analysis)|Liouville's theorem]] and the [[Little Picard theorem]]: maps from hyperbolic to parabolic to elliptic are easy, but maps from elliptic to parabolic or parabolic to hyperbolic are very constrained (indeed, generally constant!). There are inclusions of the disc in the plane in the sphere: <math>\Delta \subset \mathbf{C} \subset \widehat{\mathbf{C}},</math> but any meromorphic map from the sphere to the plane is constant, any holomorphic map from the plane into the unit disk is constant (Liouville's theorem), and in fact any holomorphic map from the plane into the plane minus two points is constant (Little Picard theorem)!
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| === Punctured spheres ===
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| These statements are clarified by considering the type of a Riemann sphere <math>\widehat{\mathbf{C}}</math> with a number of punctures. With no punctures, it is the Riemann sphere, which is elliptic. With one puncture, which can be placed at infinity, it is the complex plane, which is parabolic. With two punctures, it is the punctured plane or alternatively annulus or cylinder, which is parabolic. With three or more punctures, it is hyperbolic – compare [[pair of pants (mathematics)|pair of pants]]. One can map from one puncture to two, via the exponential map (which is entire and has an essential singularity at infinity, so not defined at infinity, and misses zero and infinity), but all maps from zero punctures to one or more, or one or two punctures to three or more are constant.
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| === Ramified covering spaces ===
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| Continuing in this vein, compact Riemann surfaces can map to surfaces of ''lower'' genus, but not to ''higher'' genus, except as constant maps. This is because holomorphic and meromorphic maps behave locally like <math>z \mapsto z^n,</math> so non-constant maps are [[ramified covering map]]s, and for compact Riemann surfaces these are constrained by the [[Riemann–Hurwitz formula]] in [[algebraic topology]], which relates the [[Euler characteristic]] of a space and a ramified cover.
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| For example, hyperbolic Riemann surfaces are [[ramified covering space]]s of the sphere (they have non-constant meromorphic functions), but the sphere does not cover or otherwise map to higher genus surfaces, except as a constant.
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| == Isometries of Riemann surfaces ==
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| The isometry group of a uniformized Riemann surface (equivalently, the conformal automorphism group) reflects its geometry:
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| * genus 0 – the isometry group of the sphere is the [[Möbius group]] of projective transforms of the complex line,
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| * the isometry group of the plane is the subgroup fixing infinity, and of the punctured plane is the subgroup leaving invariant the set containing only infinity and zero: either fixing them both, or interchanging them (1/''z'').
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| * the isometry group of the [[Poincaré half-plane model|upper half-plane]] is the real Möbius group; this is conjugate to the automorphism group of the disk.
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| * genus 1 – the isometry group of a torus is in general translations (as an [[Abelian variety]]), though the square lattice and hexagonal lattice have addition symmetries from rotation by 90° and 60°.
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| * For genus ≥ 2, the isometry group is finite, and has order at most <math>84(g-1),</math> by [[Hurwitz's automorphisms theorem]]; surfaces that realize this bound are called '''Hurwitz surfaces.'''
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| * It's known that every finite group can be realized as the full group of isometries of some riemann surface.<ref>L.Greenberg, Maximal groups and signatures, Ann. Math. Studies 79 (1974) 207–226</ref>
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| ** For genus 2 the order is maximized by the [[Bolza surface]], with order 48.
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| ** For genus 3 the order is maximized by the [[Klein quartic]], with order 168; this is the first Hurwitz surface, and its automorphism group is isomorphic to the unique simple group of order 168, which is the second-smallest non-abelian simple group. This group is isomorphic to both [[PSL(2,7)]] and [[PSL(2,7)|PSL(3,2)]].
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| ** For genus 4, [[Bring's surface]] is a highly symmetric surface.
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| ** For genus 7 the order is maximized by the [[Macbeath surface]], with order 504; this is the second Hurwitz surface, and its automorphism group is isomorphic to PSL(2,8), the fourth-smallest non-abelian simple group.
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| == Function-theoretic classification ==
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| The classification scheme above is typically used by geometers. There is a different classification for Riemann surfaces which is typically used by complex analysts. It employs a different definition for "parabolic" and "hyperbolic". In this alternative classification scheme, a Riemann surface is called ''parabolic'' if there are no nonconstant negative subharmonic functions on the surface and is otherwise called ''hyperbolic''.<ref>{{Citation | last1=Ahlfors | first1=Lars | author1-link=Lars Ahlfors | last2=Sario | first2=Leo | title=Riemann Surfaces | publisher=[[Princeton University Press]] | location=Princeton, New Jersey | edition=1st | year=1960 |page=204}}</ref><ref>{{Citation | last1=Rodin | first1=Burton | last2=Sario | first2=Leo | title=Principal Functions | publisher=[[D. Von Nostrand Company, Inc.]] | location=Princeton, New Jersey | edition=1st | year=1968 |page = 199}}</ref> This class of hyperbolic surfaces is further subdivided into subclasses according to whether function spaces other than the negative subharmonic functions are degenerate, e.g. Riemann surfaces on which all bounded holomorphic functions are constant, or on which all bounded harmonic functions are constant, or on which all positive harmonic functions are constant, etc. | |
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| To avoid confusion, call the classification based on metrics of constant curvature the ''geometric classification'', and the one based on degeneracy of function spaces ''the function-theoretic classification''. For example, the Riemann surface consisting of "all complex numbers but 0 and 1" is parabolic in the function-theoretic classification but it is hyperbolic in the geometric classification.
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| ==See also==
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| <div style="-moz-column-count:3; column-count:3;">
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| *[[Dessin d'enfant]]
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| *[[Kähler manifold]]
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| *[[Lorentz surface]]
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| *Theorems regarding Riemann surfaces
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| *[[Mapping class group]]
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| **[[branching theorem]]
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| **[[Hurwitz's automorphisms theorem]]
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| **[[identity theorem for Riemann surfaces]]
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| **[[Riemann–Roch theorem]]
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| **[[Riemann–Hurwitz formula]]
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| </div>
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{Citation | last1=Farkas | first1=Hershel M. | last2=Kra | first2=Irwin | title=Riemann Surfaces | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | isbn=978-0-387-90465-8 | year=1980}}
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| * Pablo Arés Gastesi, ''[http://www.math.tifr.res.in/~pablo/download/book/book.html Riemann Surfaces Book]''.
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| * {{Citation | last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=[[Algebraic Geometry (book)|Algebraic Geometry]] | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90244-9 | oclc=13348052 | mr=0463157 | year=1977}}, esp. chapter IV.
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| * {{Citation | last1=Jost | first1=Jürgen | title=Compact Riemann Surfaces | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-33065-3 | year=2006 | pages=208–219}}
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| *{{Citation | editor1-last=Papadopoulos | editor1-first=Athanase | title=Handbook of Teichmüller theory. Vol. I | publisher=European Mathematical Society (EMS), Zürich | series=IRMA Lectures in Mathematics and Theoretical Physics | isbn=978-3-03719-029-6 | doi=10.4171/029 | mr=2284826 | year=2007 | volume=11}}
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| *{{Citation | editor1-last=Papadopoulos | editor1-first=Athanase | title=Handbook of Teichmüller theory. Vol. II | publisher=European Mathematical Society (EMS), Zürich | series=IRMA Lectures in Mathematics and Theoretical Physics | isbn=978-3-03719-055-5 | doi=10.4171/055 | mr=2524085 | year=2009 | volume=13}}
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| *{{Citation | editor1-last=Papadopoulos | editor1-first=Athanase | title=Handbook of Teichmüller theory. Vol. III | publisher=European Mathematical Society (EMS), Zürich | series=IRMA Lectures in Mathematics and Theoretical Physics | isbn= 978-3-03719-103-3 | doi=10.4171/103 | year=2012 | volume=19}}
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| * {{Citation | last1=Siegel | first1=Carl Ludwig | author1-link=Carl Ludwig Siegel | title=Meromorphe Funktionen auf kompakten analytischen Mannigfaltigkeiten | mr=0074061 | year=1955 | journal=Nachrichten der Akademie der Wissenschaften in Göttingen. II. Mathematisch-Physikalische Klasse | issn=0065-5295 | volume=1955 | pages=71–77}}
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| *{{Citation | last1=Weyl | first1=Hermann | author1-link=Hermann Weyl | title=The concept of a Riemann surface | origyear=1913 | url=http://www.archive.org/details/dieideederrieman00weyluoft | publisher=[[Dover Publications]] | location=New York | edition=3rd | isbn=978-0-486-47004-7 | year=2009 | mr=0069903}}
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| ==External links==
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| * {{springer|title=Riemann surface|id=p/r082040}}
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| * {{planetmath reference |id=6297 | title=Riemann Surface}}
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| {{Algebraic curves navbox}}
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| {{DEFAULTSORT:Riemann Surface}}
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| [[Category:Riemann surfaces]]
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