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Riemann surface

One-dimensional complex manifold

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In mathematics, Riemann surface is German mathematician Riemann In order to give a multivalued analytic function a Single value A surface proposed by the definition domain of. In modern language, Riemann surface is a connected one dimension Complex manifold 。 The study of Riemann surface is not only one of the basic problems of the function theory of single complex variable, but also closely related to many modern mathematical branches, such as Function Theory of Multiple Complex Variables , complex manifold Algebraic geometry 、 Algebraic number theory , automorphic function, etc.

Chinese name: Riemann surface
Foreign name: Riemannian surface
Definition: Single value domain surface of multivalued analytic functions

Substantive: One-dimensional complex manifold
Presenter: Riemann
Discipline: mathematics
Discipline related: Function theory of multiple complex variables, complex manifold

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Mathematically, especially in complex analysis, a Riemannian surface is a one-dimensional Complex manifold 。 Riemann surface can be considered as a Complex plane In each point of view, they are like a complex plane, but the overall topology may be very different. For example, they can look like balls or rings, or two pages glued together.

Fig. 1 Riemannian surface with function f (z)=sqrt (z)

The point of Riemannian surface is that it can be defined between them Holomorphic function (holomorphic function)。 Riemannian surface is considered as a natural choice for studying the overall behavior of these functions, especially for those like square root and Natural logarithm In this way Multivalued function 。

Every Riemannian surface is a two-dimensional real Analytic manifold (that is, surface), but it has more structures (especially a complex structure), because the unambiguous definition of multivalued functions requires these structures. A real two-dimensional manifold can be transformed into a Riemannian surface (usually in several different ways) if and only if it is orientable. So the ball and the ring have Complex structure , but Moebius Strip ， Klein bottle And projection plane.

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Fig. 2 Riemann surface

Single value Analytic function Inverse function It can be multi valued. For example, power function and exponential function The inverse function of is Radical Functions and Logarithmic function , they are all multivalued. In addition, starting from an analytic function element, we can make Analytic development , you may get different elements in the end. Therefore, Fully analytic function It is often multivalued. When studying multivalued functions, people first decompose them into single valued analytic branches, and then connect them according to the relationship between these branches. For research, the extended complex plane is Real axis Cut, marked as functor 1, its boundary is two positive real axes, namely first quadrant The lower and Delta Quadrant Order on Function 1

Riemann surface

You get one Single value Analytic branch, which is analytic in the interior of func 1, and continues to the boundary agent Positive real number x The corresponding ones are located at two points on the agent and ring respectively, but take different values. Set function 2 is another positive edge Real axis The boundary of a secant extended complex plane is denoted as 崹 and 崹. Another single value resolution branch obtained by setting. Different from, on Positive real number x At the corresponding two points, the values are respectively. Since the values on the agent and the element are the same as those on the element and the element respectively, people naturally bond the agent and the element and the element together, and splice the function 1 and the function 2 into a whole, which is called Riemann surface. As a function defined on the surface, it contains two branches Single value Of. for Multivalued function It is Riemann's original idea to construct an appropriate definition place and make it a complete single valued analytic function. Constructed in this way, and ln z The Riemannian surface of is shown in Figure 1.

Riemann surface atlas (1 sheet)

Put the Riemannian surface on the extended complex plane according to its original position, and it becomes one of the extended complex planes n Leaf coverage surfaces. Point on surface O And ∞ are called n -Level 1 branch point. Similarly, ln z The Riemannian surface of is the covering surface of the complex plane without branch points (after removing the origin). Generally speaking, any covering surface of a complex plane (or extended complex plane) can be regarded as a Riemannian surface. Set points in the coverage surface P Point in complex plane z Above is called z by P Projection of. Whether a function defined on the surface is resolved at non branch points depends on its projection z Whether the function of is analytic; When the projection is z 0's n -At the level 1 branch point, it depends on whether it is analytic. This is Riemann's original concept of Riemann surface. The classical theory of Riemannian surfaces is developed from this concept.

Fig. 3 Riemann surface

A completely analytic function or configuration, in which z The function element centered on 0 is regarded as placed in z The point on 0 naturally becomes the covering surface of the extended plane, which is its Riemannian surface. One Algebra function w = w ( z ）Riemannian surface of is an extended plane n Leaf coverage surface（ n In the corresponding equation w The highest number of times). For example, the construction of Riemannian surface of is shown in Figure 1. Cut the two line segments connecting 0,1 and 2,3 in the upper and lower planes into cracks. Each crack produces two sides, which are respectively connected to the upper and lower half of the plane, and are represented by solid lines and dotted lines. Then glue the edge shown by the solid line (dashed line) in the upper plane and the edge shown by the dashed line (solid line) in the lower plane.

(C.H.) H. Weir Firstly, the modern definition of Riemannian surface is given. At the same time, he also gave a strict definition of "manifold", the basic concept of modern mathematics. According to Weier, Riemannian surface is a one-dimensional complex manifold. On a surface (a connected Hausdorff space which is locally homeomorphic to the Euclidean plane), a family of local parameters (a Open set A continuous univalent complex value function, also called local coordinates, on the common part of the definition domain of any two adjacent local parameters, if one parameter is resolved as a function of the other parameter, and the definition domain of these parameters covers the entire surface, then the surface together with this family of local parameters (called conformal structure) It forms a Riemannian surface. Complex plane C perhaps C Any region above is Riemannian surface according to its natural parameters. In the extended complex plane function, except in the C In addition to the existing natural parameter{ z ││ z │>0} (including Infinity point ）Let the function become a Riemannian surface. The continuous mapping from a Riemannian surface to a Riemannian surface is called analytic analytic function 。 The analytic mapping from a Riemannian surface to Fuli is Semipure function （ Meromorphic function ）。 Harmonic (or subharmonic) functions on Riemannian surfaces are defined as functions whose local parameters are harmonic (or subharmonic). The introduction of Riemann surface greatly expanded Complex function theory Scope of study.

A Riemannian surface made of compact surfaces is called a closed Riemannian surface, otherwise it is called a open Riemannian surface. If a Closed surface The rank of one dimensional homology group (or one dimensional homology group of module ideal boundary) on (or open surface) is 2 g , then g (non negative integer or infinite) Genus 。 The genus of an open surface may be infinite. Two Riemannian surfaces are conformally equivalent, if there is a one-to-one analytic mapping from one surface to another（ Conformal mapping ）。 The same genus g ( g >1) All conformals of closed Riemannian surfaces of Equivalence class The so-called Modular space 。 Riemann first found that the elements in the module space are composed of 3 g -Three complex parameters are determined. From the study of modular space, a variety of Teschmiller space The theory of.

Fig. 4 Riemann surface

People also classified the kelvin surface. The open surfaces without non constant negative subharmonic functions are called parabolic surfaces, and other open surfaces are called hyperbolic surfaces. The application of parabolic surface O G means. There is no bounded analytic or harmonic function of non constant, Dirichlet integral The open surfaces of finite analytic or harmonic functions, or positive harmonic functions, respectively form classes O AB or O HB， O AD or O HD, or O HP。 Among these surface classes, there is an inclusion relationship as follows: according to Riemann's original concept, Riemann surfaces are functional covering surfaces. The so-called surface 愞 is a surface F The covering surface of refers to the existing surface 愞 To surface F Mapping in ƒ , for each support ∈ 愞, both have A and ƒ (A) ∈ F Open neighbor sum of V , so that the limits and V between, ƒ Topology is equivalent to the mapping of the unit circle to itself z = z N (n is a positive integer, which is related to A; When n>1, A is called branch point). The mapping in the definition is called projection. When F is a Riemannian surface, the local parameters of F can be made. Let z be a local parameter of 愞, which defines a conformal structure on 愞 and makes it a Riemannian surface, and ƒ is an analytic mapping. The Riemannian surface of a completely analytic function w=g (z) is the covering surface of the function, and is given a conformal structure according to the above method. There are two semipure functions on this surface: take w=g (z) as a single valued function on the surface, and write w=G (P); There is also the projection from the surface to the function, marked with z=Z (P), P is the point on the surface. The complete analytic function here can include polar elements, branching elements, and branched polar elements. Two curves (continuous curves) with the same starting point and the same ending point on a surface are called homotopy: t → φ i (t) (0 ≤ t ≤ 1, i=1,2). If there is a continuous mapping (t, u) → φ (t, u) (0 ≤ t ≤ 1,0 ≤ u ≤ 1) to the surface, so that φ (t, 0) 呏 φ 1 (t), φ (t, 1) 呏 φ 2 (t), φ (0, u) 呏 φ 1 (0), φ (1, u) 呏 3 φ 1 (1). All homotopy equivalence classes composed of closed curves with fixed endpoints on the surface form a group with the connection of curves as multiplication operation, which is called the basic group of the surface about this fixed point. Basic groups about different points are isomorphic to each other. A surface that contains only one element of a primitive group is called a simply connected surface. A covering surface without branches is called a smooth covering surface. Let ƒ make 愞 a smooth covering surface of F. If у=ƒ (), where and у are curves on 愞 and F respectively, it is called the lifting of у. If for any ° F and any ° A ° projected from the starting point of ° F, the elevation of ° A ° always exists, then ° F is the normal covering surface of F. Smoothness guarantees the uniqueness of the specified starting point. Singularity theorem says that if 愞 is a normal covering surface of F, then for any two mutually homotopy curves v1 and v2 on F, and any point A in 愞 projected from the common starting point of v1 and v2, the lifting and 2 of v1 and у 2 starting from A always have a common end point, and 1 and 2 are also homotopy (on 愞). The single value theorem of analytic function elements along curves in complex function theory is a specific application of this theorem. A simply connected normal covering surface is called a universal covering surface. For any surface F, its universal covering surface 愞 always exists and is unique in the sense of conformal equivalence. When F is a Riemannian surface, 愞 can also become a Riemannian surface, and projection ƒ is an analytic mapping. famous Monovalued theorem It is said that a simply connected Riemannian surface must be conformal equivalent to a function (closed), C (parabolic) or unit circle (hyperbolic). If 愞=function, then F=function. If 愞=C, then F=C, C {0}, or torus (torus is a closed surface with genus 1; conversely, the universal covering (Riemann) surface of torus must be C). When 愞 is a unit circle, all satisfy ƒ. The conformal mapping φ (called covering transformation) of φ=ƒ forms a Fuchs group. Therefore, except for the above special exceptions, every Riemannian surface can be expressed as the quotient of the unit circle with respect to a Fuchs group; Therefore, Fractional linear transformation The theory of discontinuous groups (Klein group, including Fuchs group) is closely related to Riemann surface theory. If F here is a Riemannian surface of completely analytic function w=g (z), then G (ƒ (t)) and Z (ƒ (t)) (t ∈ function, C, or unit circle) are semi pure functions, and the multivalued function w=g (z) is univalued by parameter t (called univalued parameter). Thus, the famous Hilbert's 22nd problem, namely, the problem of singularization, is solved. On a Riemannian surface, if a differential ƒ (z) dz is defined for each local parameter z (ƒ (z) is a semi pure function), and the corresponding ƒ (z) dz and φ (ζ) d ζ of the two adjacent parameters z and ζ meet the relationship ƒ (z (ζ)) · z ┡ (ζ)=φ (ζ), then a semi pure differential is defined on the surface. The order of the zero point or pole of a semi pure function (or semi pure differential) at a certain point is equal to the order of the zero point and pole of the function (or the coefficient of the differential in the representation form under this parameter) as the local parameter at the point after a local parameter is taken. Riemann Roch theorem states that: on a closed surface with genus g, the points p1, p2,..., ps; q1, q2,..., qt and the positive integer k1 are specified, k2,…,ks;n1,n2,…,nt, Order. Let the dimension of a linear space on a complex number field consisting of all semipure functions (or semipure differentials) with pi as the pole of at most ki order (or at least the zero of ki order, i=1,2,..., s) and qi as the zero of at least ni order (or at most the pole of ni order, i=1,2,..., t) be A (or B), then A=B+m-g+1. This theorem is a basic result of the theory of closed Riemannian surfaces; Under certain conditions, it is also extended to open surfaces and high-dimensional complex manifolds.^[1]

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The geometric properties of Riemannian surfaces are the most wonderful. They also go to other curves, manifold or Algebraic variety The promotion on provides intuitive understanding and motivation. Riemann Roch theorem This is the best example of this impact.

Make X a Hausdorff space (Hausdorff space)。 A subset from an open subset U ⊂ X to C Homeomorphism It is called chart Two graphs f and g with overlapping regions are compatible if the mappings f o g-1 and g o f-1 are holomorphic in the domain. If A is a group of compatible graphs, and x in each X is in the definition domain of some f, then A is called an atlas. When we give X a graph set A, we call (X, A) a Riemannian surface. If we know there is a atlas, we call X Riemannian surface for short.

Different atlases can give essentially the same Riemannian surface structure on X; In order to avoid this ambiguity, we sometimes require X to be maximal, that is, it is not in any larger atlas subset 。 According to Zorn's Lemma, each graph set A is contained in a unique maximum graph set.

The complex plane C is probably the most ordinary Riemannian surface. Map f (z)=z（ Identity mapping ）It defines a graph of C, but a graph set of C Map g (z)=z *（ conjugate ）Mapping also defines a graph of C and a graph set of C Figures f and g are incompatible, so they each give C a Riemannian surface structure. In fact, given Riemannian surface X and its graph set A, the conjugate graph set B={f *: f ∈ A} is always incompatible with A, so X is given a different Riemannian surface structure.

Similarly, the open of each complex plane subset It can be regarded as Riemannian surface naturally. More generally, the open subset of every Riemannian surface is a Riemannian surface.

Let S=C →{∞} and let f (z)=z where z belongs to S {∞} and let g (z)=1/z where z belongs to S and define 1/∞ as 0 Then f and g are graphs, they are compatible, and {f, g} is a set of S graphs, making S a Riemannian surface. This special surface is called Li Man Ball Because it can be interpreted as wrapping the complex plane on a ball. Unlike the complex plane, it is a compact space.^[2]

Escher's Gallery also uses Riemann surface.

Compact Riemannian surfaces can be regarded as and defined in complex Nonsingularity on Algebraic curve Equivalent. Important examples of noncompact Riemannian surfaces are given by analytical continuity

The function f: M → N between two Riemannian surfaces M and N is called holomorphic. If for each graph g in the graph set of M and each graph h in the graph set of N, the mapping h o f o g - 1 is holomorphic in all defined places (as a function from C to C). The composition of two holomorphic functions is holomorphic. Two Riemannian surfaces M and N are called conformally equivalent Birefringence The holomorphic function from M to N and its inverse are also holomorphic (the last condition is automatically met, so it can be omitted). Two Riemannian surfaces with conformal equivalence are identical for all practical applications.

each simply-connected Riemannian surface and C or Li Man Ball C →{∞} or open disk {z ∈ C: | z |<1} Conformal Equivalence. This proposition is called the consistency theorem.

each connected Riemannian surface can be transformed into a constant curvature -Of 1, 0 or 1 complete real Riemannian manifold 。 In addition to the Riemannian structure measure Is unique. Riemannian surface with curvature - 1 is called hyperbolic; The open disk is a classic example. A Riemannian surface with curvature 0 is called parabolic; C is a typical parabolic Riemannian surface. Finally, a Riemannian surface with curvature+1 is called elliptical; Li Man Ball C →{∞} is an example of this

For each closed parabolic Riemannian surface, Fundamental group It is isomorphic to lattice groups of order 2, so the surface can be constructed as C/Γ, where C is a complex plane and Γ is a lattice group. Coset The set represented by is called the basic field.^[3]

Similarly, for each hyperbolic Riemannian surface, the basic Group isomorphism In the Fuchsian group, so the surface can be constructed by the Fuchsian model H/Γ, where H is the upper Half plane And Γ is a Fuchsian group. The representative of H/Γ coset is a free regular set, which can be used as a metric basic polygon.

When a hyperbolic surface is compact, the total area of the surface is 4 pi (g-1), where g is the genus of the surface; The area can be calculated by applying Gauss Bonnet theorem to the area of basic polygons.

We mentioned earlier that Riemannian surfaces, like all complex manifolds, are like real manifolds Orientable 。 Because complex graphs f and g have transformation functions h=f (g-1 (z)), we can think that h is a mapping from R2 open set to R2, and the Jacobian matrix at point z is a real linear transformation given by the operation of multiplying the complex number h '(z). But, multiply by the complex number α determinant Is equal to | α | ^ 2, so the Jacobian matrix of h has a positive determinant value. Therefore, complex graph set is orientable graph set.

Riemann He was the first to study Riemannian surfaces. Riemann surface is named after him.

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Riemann (G.F.B Riemann) was born in Bresselentz, Hanover, Germany, on September 17, 1826, and died in Silasca, Italy, on July 20, 1866.

Riemann is right modern mathematics As one of the most influential mathematicians, we can see from his mathematical level at that time that his achievements as a great analyst can be divided into eight areas. The first four fields are about complex analysis. He was the first to consciously transition the real field to the complex field, creating the complex variable function field, Algebra Function theory, Analytic Theory of Ordinary Differential Equations And analytic number theory; The last four fields mainly involve real analysis, including integral theory, triangular theory, Differential geometry Major breakthroughs have been made in mathematical and physical equations. What's important is that the achievements more than a century ago are directly the same as those in modern mathematics topology Method, general manifold concept, Riemann Loch theorem connecting topology and analysis, Algebraic geometry In particular, Abelian clusters and parameter modules are closely connected Space concept And Riemannian geometry is more indicative of general relativity, and it is he who initiated the revolutionary transformation of modern mathematics.

Riemann, the son of a priest, studied at Gottingen University and Merlin University. In 1851, he obtained a doctor's degree at Gottingen University. In 1854, he served as a part-time lecturer at the university. In 1857, he served as an associate professor. In 1859, he served as P G. Professor Ren, successor of L. Dirichlet. He died young because of lung disease. In his short life, he has made epoch-making contributions in various fields of mathematics. The most important contributions are in four aspects: geometry, complex variable function theory, differential equation and basic theory of mathematical analysis. He is the founder of Riemannian geometry and one of the founders of complex function theory. In mathematical analysis, his standard definition of integral has been used so far that the classical integral in this sense is called "Riemann integral". He's also right Fourier series Many researches have been done on the theory, the most famous of which is the theorem named after him. Riemann also created some important methods for the theory of partial differential equations and ordinary differential equations, especially the singularity theory of ordinary differential equations. Riemann also paid close attention to natural science, especially physics. His research on complex variable functions and differential equations is directly related to fluid mechanics and electromagnetic theory. The famous mathematician Klein once pointed out in the "Lectures on the Development of Mathematics in the 19th Century" that "Riemann used his mathematical talents to open up new ways for natural science itself. Then he took natural science as the power to form new concepts in mathematics".^[4]

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