Хэрэглэгч:Timur/Ноорог/Риман гадаргуу

Riemann surface for the function f(z) = sqrt(z)

In mathematics, particularly in complex analysis, a Riemann surface, 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.

The main point of Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi-valued functions such as the square root or the logarithm.

Every Riemann surface is a two-dimensional real analytic manifold (i.e., a surface), but it contains more structure (specifically a 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. So the sphere and torus admit complex structures, but the Möbius strip, Klein bottle and projective plane do not.

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.

Формаль тодорхойлолт

Let X be a Hausdorff space. A homeomorphism from an open subset U⊂X to a subset of C is called a chart. Two charts f and g whose domains intersect are said to be compatible if the maps f o g⁻¹ and g o f ⁻¹ are holomorphic over their domains. If A is a collection of compatible charts and if any x in X is in the domain of some f in A, then we say that A is an atlas. When we endow X with an atlas A, we say that (X, A) is a Riemann surface. If the atlas is understood, we simply say that X is a Riemann surface.

Different atlases can give rise to essentially the same Riemann surface structure on X; to avoid this ambiguity, one sometimes demands that the given atlas on X be maximal, in the sense that it is not contained in any other atlas. Every atlas A is contained in a unique maximal one by Zorn's lemma.

Жишээнүүд

• The complex plane C is perhaps the most basic Riemann surface. The map f(z) = z (the identity map) defines a chart for C, and {f} is an atlas for C. The map g(z) = z^* (the 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^* : f ∈ A} is never compatible with A, and endows X with a distinct, incompatible Riemann structure.

• 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.

• 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.

• The theory of compact Riemann surfaces can be shown to be equivalent to that of projective algebraic curves that are defined over the complex numbers and non-singular. Important examples of non-compact Riemann surfaces are provided by analytic continuation (see below.)

Шинж чанарууд ба бусад тодорхойлолтууд

A function f: M → N between two Riemann surfaces M and N is called holomorphic if for every chart g in the atlas of M and every chart h in the atlas of N, the map h o f o g^-1 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 conformally equivalent 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.

Every simply connected Riemann surface is conformally equivalent to C or to the Riemann sphere C ∪ {∞} or to the open disk {z ∈ C : |z| < 1}. This is the main step in the uniformization theorem.

The uniformization theorem states that every connected Riemann surface admits a unique complete 2-dimensional real Riemann metric with constant curvature -1, 0 or 1. The Riemann surfaces with curvature -1 are called hyperbolic; the open disk with the Poincaré-metric of constant curvature -1 is the local model. Examples include all surfaces with genus g>1. The Riemann surfaces with curvature 0 are called parabolic; C and the torus are typical parabolic Riemann surfaces. Finally, the surfaces with curvature +1 are known as elliptic; the Riemann sphere C ∪ {∞} is the only example.

For every closed parabolic Riemann surface, the fundamental group is isomorphic to a rank 2 lattice, and thus the surface can be constructed as C/Γ, where C is the complex plane and Γ is the lattice. The set of representatives of the cosets are called fundamental domains.

Similarly, 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 sets and can be fashioned into metric fundamental polygons.

When a hyperbolic surface is compact, then the total area of the surface is 4π(g-1), where g is the genus of the surface; the area is obtained by applying the Gauss-Bonnet theorem to the area of the fundamental polygon.

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^-1(z)) we can consider h as a map from an open set of R² to R² whose Jacobian in a point z is just the real linear map given by multiplication by the complex number h'(z). However, the real determinant of multiplication by a complex number α equals |α|², so the Jacobian of h has positive determinant. Consequently the complex atlas is an oriented atlas.

Бусад жишээнүүд

• As noted above, the Riemann sphere is the only elliptic Riemann surface.

• The only parabolic, simply connected Riemann surface is the complex plane. All parabolic surfaces can be obtained as a quotient of the plane. All parabolic surfaces are homeomorphic to either the plane, the annulus, or the torus. However it does not follow that all tori are biholomorphic to each other. This is the first appearance of the problem of moduli. The modulus of a toris can be captured by a single complex number with positive imaginary part. In fact, the marked moduli space (Teichmüller space) of the torus is biholomorphic to the open unit disk.

• The only hyperbolic, simply connected Riemann surface is the open unit disk. The celebrated Riemann Mapping Theorem states that any simply connected strict subset of the complex plane is biholomorphic to the unit disk. All hyperbolic surfaces are quotients of the unit disk. 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 with finitely generated fundamental group. Any one of these has a finite number of moduli and so a finite dimensional Teichmüller space. The 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, 3g - 3 complex parameters suffice.

Функцүүд

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.

In contrast, on a compact Riemann surface 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 ∪ {∞}).

Түүх

Riemann surfaces were first studied by Bernhard Riemann and were named after him.

Урлаг уран зохиолд

• One of M.C. Escher's works, Print Gallery, is laid out on a cyclically growing grid that has been described as a Riemann surface.
• In Aldous Huxley's novel Brave New World, "Riemann Surface Tennis" is a popular game.

Мөн үзэх

• Algebraic geometry
• Conformal geometry
• Kähler manifold
• Teichmüller space
• Riemann sphere
• Theorems regarding Riemann surfaces
  • branching theorem
  • Hurwitz's automorphisms theorem
  • identity theorem for Riemann surfaces
  • Riemann-Roch theorem
  • Riemann-Hurwitz formula
  • Riemann mapping theorem
  • uniformization theorem

Ном хэвлэл

• Hershel M. Farkas and Irwin Kra, Riemann Surfaces (1980), Springer-Verlag, New York. ISBN 0-387-90465-4
• Jürgen Jost, Compact Riemann Surfaces (2002), Springer-Verlag, New York. ISBN 3-540-43299-X
• Pablo Arés Gastesi, Riemann Surfaces Book.