Риман беттерінің фундаментальды көпбұрыштары

In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0. It encodes not only the topology of the surface through its fundamental group but also determines the Riemann surface up to conformal equivalence. By the uniformization theorem, every compact Riemann surface has simply connected universal covering surface given by exactly one of the following:

the Riemann sphere,
the complex plane,
the unit disk D or equivalently the upper half plane H.

In the first case of genus zero, the surface is conformally equivalent to the Riemann sphere. In the second case of genus one, the surface is conformally equivalent to a torus C/Λ for some lattice Λ in C. The fundamental polygon of Λ, if assumed convex, may be taken to be either a period parallelogram or a centrally symmetric hexagon, a result first proved by Fedorov in 1891. In the last case of genus g > 1, the Riemann surface is conformally equivalent to H/Γ where Γ is a Fuchsian group of Möbius transformations. A fundamental domain for Γ is given by a convex polygon for the hyperbolic metric on H. These can be defined by Dirichlet polygons and have an even number of sides. The structure of the fundamental group Γ can be read off from such a polygon. Using the theory of quasiconformal mappings and the Beltrami equation, it can be shown there is a canonical convex fundamental polygon with 4g sides, first defined by Fricke, which corresponds to the standard presentation of Γ as the group with 2g generators a1, b1, a2, b2, , ag, bg and the single relation [a1,b1][a2,b2] ⋅⋅⋅ [ag,bg] = 1, where [a,b] = a b a−1b−1. Any Riemannian metric on an oriented closed 2 manifold M defines a complex structure on M, making M a compact Riemann surface. Through the use of fundamental polygons, it follows that two oriented closed 2 manifolds are classified by their genus, that is half the rank of the Abelian group Γ/[Γ,Γ], where Γ = 1(M). Moreover, it also follows from the theory of quasiconformal mappings that two compact Riemann surfaces are diffeomorphic if and only if they are homeomorphic. Consequently, two closed oriented 2 manifolds are
homeomorphic if and only if they are diffeomorphic. Such a result can also be proved using the methods of differential topology.

In the case of genus one, a fundamental convex polygon is sought for the action by translation of Λ = Z a ⊕ Z b on R2 = C where a and b are linearly independent over R. (After performing a real linear transformation on R2, it can be assumed if necessary that Λ = Z2 = Z + Z i; for a genus one Riemann surface it can be taken to have the form Λ = Z2 = Z + Z ω, with Im ω > 0.) A fundamental domain is given by the parallelogram s 'x' + t 'y' for 1=0 < s , t < 1 where 'x' and 'y' are generators of Λ. If C is the interior of a fundamental convex polygon, then the translates + x cover R2 as x runs over Λ. It follows that the boundary points of C are formed of intersections ∩ ( + x). These are compact convex sets in ∂C and thus either vertices of C or sides of C. It follows that every closed side of C can be written this way. Translating by −x it follows that ∩ ( − x) is also a side of C. Thus sides of C occur in parallel pairs of equal length. The end points of two such parallel segments of equal length can be joined so that they intersect and the intersection occurs at the midpoints of the line segments joining the endpoints. It follows that the intersections of al such segments occur at the same point. Translating that point to the origin, it follows that the polygon is centrally symmetric; that is, if a point z is in the polygon, so too is −z. It is easy to see translates of a centrally symmetric convex hexagon tessellate the plane. If A is a point of the hexagon, then the lattice is generated by the displacement vectors AB and AC where B and C are the two vertices which are not neighbours of A and not opposite A. Indeed, the second picture shows how the hexagon is equivalent to the parallelogram obtained by displacing the two triangles chopped off by the segments AB and AC. Equally well the first picture shows another way of matching a tiling by parallelograms with the hexagonal tiling. If the centre of the hexagon is 0 and the vertices in order are a, b, c, −a, −b and −c, then Λ is the Abelian group with generators 'a' + 'b' and 'b' + 'c'.

For a lattice Λ in C = R2, a fundamental domain can be defined canonically using the conformal structure of C. Note that the group of conformal transformations of C is given by complex affine transformations 1=g(z) = az + b with a ≠ 0. These transformations preserve Euclidean metric up to a factor, as well as preserving the orientation. It is the subgroup of the Möbius group fixing the point at ∞. The metric structure can be used to define a canonical fundamental domain by 1=C = {z: d(z, 0) < d(z, λ) for all λ ≠ 0 in Λ}. (It is obvious from the definition that it is a fundamental domain.) This is an example of a Dirichlet domain or Voronoi diagram: since complex translations form an Abelian group, so commute with the action of Λ, these concepts coincide. The canonical fundamental domain for 1=Λ = Z + Zω with Im ω > 0 is either a symmetric convex parallelogram or hexagon with centre 0. By conformal equivalence, the period ω can be further restricted to satisfy and As Dirichlet showed ("Dirichlet's hexagon theorem", 1850), for almost all ω the fundamental domain is a hexagon. For Re ω > 0, the midpoints of sides are given by ±1/2, ±ω/2 and ±(ω – 1)/2; the sides bisect the corresponding radii from 0 orthogonally, which determines the vertices completely. In fact the first vertex must have the form (1 + ix)/2 and ω(1 + iy)/2 with x and y real; so if 1=ω = a + ib, then 1=a – by = 1 and 1=x = b + ay. Hence 1=y = (a – 1)/b and 1=x = (a2 + b2 – a)/b. The six vertices are therefore ±ω(1 – iy)/2 and ±(1 ± ix)/2.

In this section, starting from an arbitrary Dirichlet polygon, a description will be given of the method of , elaborated in , for modifying the polygon to a non convex polygon with 4g equivalent vertices and a canonical pairing on the sides. This treatment is an analytic counterpart of the classical topological classification of orientable 2 dimensional polyhedra presented in .

Given a Riemann surface of genus g greater than one, Fricke described another fundamental polygon, the Fricke canonical polygon, which is a very special example of a Dirichlet polygon. The polygon is related to the standard presentation of the fundamental group of the surface. Fricke's original construction is complicated and described in Using the theory of quasiconformal mappings of Ahlfors and Bers, gave a new, shorter and more precise version of Fricke's construction. The Fricke canonical polygon has the following properties:

The vertices of the Fricke polygon has 4g vertices which all lie in an orbit of Γ. By vertex is meant the point where two sides meet. The sides are matched in distinct pairs, so that there is a unique element of Γ carrying a side to the paired side, reversing the orientation. Since the action of Γ is orientation preserving, if one side is called , then the other of the pair can be marked with the opposite orientation The edges of the standard polygon can be arranged so that the list of adjacent sides takes the form That is, pairs of sides can be arranged so that they interleave in this way. The sides are geodesic arcs. Each of the interior angles of the Fricke polygon is strictly less than , so that the polygon is strictly convex, and the sum of these interior angles is 2. The above construction is sufficient to guarantee that each side of the polygon is a closed (non trivial) loop in the Riemann surface H/Γ. As such, each side can thus an element of the fundamental group In particular, the fundamental group has 2g generators , with exactly one defining constraint,

The genus of the Riemann surface H/Γ is g.

The area of the standard fundamental polygon is where g is the genus of the Riemann surface (equivalently, where 4g is the number of the sides of the polygon). Since the standard polygon is a representative of H/Γ, the total area of the Riemann surface is equal to the area of the standard polygon. The area formula follows from the Gauss–Bonnet theorem and is in a certain sense generalized through the Riemann–Hurwitz formula.