1.
Riemann surface
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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 plane, locally near every point they look like patches of the complex plane. For example, they can look like a sphere or a torus or several sheets glued together, the main point of Riemann surfaces is that holomorphic functions may be defined between them. Every Riemann surface is a real analytic manifold, but it contains more structure which is needed for the unambiguous definition of holomorphic functions. A two-dimensional real manifold can be turned into a Riemann surface if, 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 provide the intuition and motivation for generalizations to other curves. The Riemann–Roch theorem is an example of this influence. There are several equivalent definitions of a Riemann surface, a Riemann surface X is a complex manifold of complex dimension one. This means that X is a Hausdorff topological space endowed with an atlas, the map carrying the structure of the complex plane to the Riemann surface is called a chart. Additionally, the maps between two overlapping charts are required to be holomorphic. A Riemann surface is a manifold of dimension two – a two-sided surface – together with a conformal structure. Again, manifold means that locally at any point x of X, 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 metrics. Two such metrics are considered 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. A complex structure gives rise to a structure by choosing the standard Euclidean metric given on the complex plane. Showing that a structure determines a complex structure is more difficult. The complex plane C is the most basic Riemann surface, the map f = z defines a chart for C, and is an atlas for C. The map g = z* also defines a chart on C and 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 = is never compatible with A, in an analogous fashion, every non-empty open subset of the complex plane can be viewed as a Riemann surface in a natural way
2.
Hyperbolic geometry
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In mathematics, hyperbolic geometry is a non-Euclidean geometry. Hyperbolic plane geometry is also the geometry of saddle surface or pseudospherical surfaces, surfaces with a constant negative Gaussian curvature, a modern use of hyperbolic geometry is in the theory of special relativity, particularly Minkowski spacetime and gyrovector space. In Russia it is commonly called Lobachevskian geometry, named one of its discoverers. This page is mainly about the 2-dimensional hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry, Hyperbolic geometry can be extended to three and more dimensions, see hyperbolic space for more on the three and higher dimensional cases. Hyperbolic geometry is closely related to Euclidean geometry than it seems. When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry, there are two kinds of absolute geometry, Euclidean and hyperbolic. All theorems of geometry, including the first 28 propositions of book one of Euclids Elements, are valid in Euclidean. Propositions 27 and 28 of Book One of Euclids Elements prove the existence of parallel/non-intersecting lines and this difference also has many consequences, concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry, new concepts need to be introduced. Further, because of the angle of parallelism hyperbolic geometry has an absolute scale, single lines in hyperbolic geometry have exactly the same properties as single straight lines in Euclidean geometry. For example, two points define a line, and lines can be infinitely extended. Two intersecting lines have the properties as two intersecting lines in Euclidean geometry. For example, two lines can intersect in no more than one point, intersecting lines have equal opposite angles, when we add a third line then there are properties of intersecting lines that differ from intersecting lines in Euclidean geometry. For example, given 2 intersecting lines there are many lines that do not intersect either of the given lines. While in some models lines look different they do have these properties, non-intersecting lines in hyperbolic geometry also have properties that differ from non-intersecting lines in Euclidean geometry, For any line R and any point P which does not lie on R. In the plane containing line R and point P there are at least two lines through P that do not intersect R. This implies that there are through P an infinite number of lines that do not intersect R. All other non-intersecting lines have a point of distance and diverge from both sides of that point, and are called ultraparallel, diverging parallel or sometimes non-intersecting. Some geometers simply use parallel lines instead of limiting parallel lines and these limiting parallels make an angle θ with PB, this angle depends only on the Gaussian curvature of the plane and the distance PB and is called the angle of parallelism
3.
Hurwitz automorphisms theorem
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A group for which the maximum is achieved is called a Hurwitz group, and the corresponding Riemann surface a Hurwitz surface. Because compact Riemann surfaces are synonymous with non-singular complex projective algebraic curves, the theorem is named after Adolf Hurwitz, who proved it in. For example the double cover of the projective line y2 = xp −x branched at all points defined over the field has genus g=/2 but is acted on by the group SL2 of order p3−p. One of the themes in differential geometry is a trichotomy between the Riemannian manifolds of positive, zero, and negative curvature K. It manifests itself in diverse situations and on several levels. While in the first two cases the surface X admits infinitely many automorphisms, a hyperbolic Riemann surface only admits a discrete set of automorphisms. By the uniformization theorem, any hyperbolic surface X – i. e. the Gaussian curvature of X is equal to one at every point – is covered by the hyperbolic plane. The conformal mappings of the surface correspond to orientation-preserving automorphisms of the hyperbolic plane, by the Gauss–Bonnet theorem, the area of the surface is A = − 2π χ = 4π. In order to make the automorphism group G of X as large as possible, we want the area of its fundamental domain D for this action to be as small as possible. If the fundamental domain is a triangle with the vertex angles π/p, π/q and π/r, defining a tiling of the plane, then p, q, and r are integers greater than one. Thus we are asking for integers which make the expression 1 − 1/p − 1/q − 1/r strictly positive and this minimal value is 1/42, and 1 − 1/2 − 1/3 − 1/7 = 1/42 gives a unique triple of such integers. This would indicate that the order |G| of the group is bounded by A/A ≤168. However, a more delicate reasoning shows that this is an overestimate by the factor of two, because the group G can contain orientation-reversing transformations, for the orientation-preserving conformal automorphisms the bound is 84. To obtain an example of a Hurwitz group, let us start with a -tiling of the hyperbolic plane and its full symmetry group is the full triangle group generated by the reflections across the sides of a single fundamental triangle with the angles π/2, π/3 and π/7. Since a reflection flips the triangle and changes the orientation, we can join the triangles in pairs, a Hurwitz surface is obtained by closing up a part of this infinite tiling of the hyperbolic plane to a compact Riemann surface of genus g. This will necessarily involve exactly 84 double triangle tiles, note that the polygons in the tiling are not fundamental domains – the tiling by triangles refines both of these and is not regular. Wythoff constructions yields further uniform tilings, yielding eight uniform tilings and these all descend to Hurwitz surfaces, yielding tilings of the surfaces. This is the last part of the theorem of Hurwitz, the smallest Hurwitz group is the projective special linear group PSL, of order 168, and the corresponding curve is the Klein quartic curve
4.
Schwarz triangle
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In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere, possibly overlapping, through reflections in its edges. These can be defined generally as tessellations of the sphere. Each Schwarz triangle on a sphere defines a group, while on the Euclidean or hyperbolic plane they define an infinite group. A Schwarz triangle is represented by three rational numbers each representing the angle at a vertex, the value n/d means the vertex angle is d/n of the half-circle. When these are numbers, the triangle is called a Möbius triangle, and corresponds to a non-overlapping tiling. A Schwarz triangle is represented graphically by a triangular graph, each node represents an edge of the Schwarz triangle. Each edge is labeled by a value corresponding to the reflection order. Order-2 edges represent perpendicular mirrors that can be ignored in this diagram, the Coxeter-Dynkin diagram represents this triangular graph with order-2 edges hidden. A Coxeter group can be used for a simpler notation, as for graphs, and = for. Density 10, The Schwarz triangle is the smallest hyperbolic Schwarz triangle and its triangle group is the triangle group, which is the universal group for all Hurwitz groups – maximal groups of isometries of Riemann surfaces. All Hurwitz groups are quotients of the group, and all Hurwitz surfaces are tiled by the Schwarz triangle. The smallest Hurwitz group is the group of order 168, the second smallest non-abelian simple group, which is isomorphic to PSL. The triangle tiles the Bolza surface, a highly symmetric surface of genus 2, the triangles with one noninteger angle, listed above, were first classified by Anthony W. Knapp in. A list of triangles with multiple noninteger angles is given in, 3D The general Schwarz triangle and the generalized incidence matrices of the corresponding polyhedra
5.
Hurwitz surface
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In Riemann surface theory and hyperbolic geometry, a Hurwitz surface, named after Adolf Hurwitz, is a compact Riemann surface with precisely 84 automorphisms, where g is the genus of the surface. This number is maximal by virtue of Hurwitzs theorem on automorphisms and they are also referred to as Hurwitz curves, interpreting them as complex algebraic curves. The Fuchsian group of a Hurwitz surface is a finite index torsionfree normal subgroup of the triangle group, the finite quotient group is precisely the automorphism group. The group of complex automorphisms is a quotient of the triangle group. The Hurwitz surface of least genus is the Klein quartic of genus 3, with group the projective special linear group PSL, of order 84 =168 = 22·3·7. An interesting phenomenon occurs in the possible genus, namely 14. Here there is a triple of distinct Riemann surfaces with the automorphism group. The explanation for this phenomenon is arithmetic, namely, in the ring of integers of the appropriate number field, the rational prime 13 splits as a product of three distinct prime ideals. The principal congruence subgroups defined by the triplet of primes produce Fuchsian groups corresponding to the first Hurwitz triplet
6.
Klein quartic
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As such, the Klein quartic is the Hurwitz surface of lowest possible genus, see Hurwitzs automorphisms theorem. Its automorphism group is isomorphic to PSL, the second-smallest non-abelian simple group, the quartic was first described in. Originally, the Klein quartic referred specifically to the subset of the projective plane P2 defined by an algebraic equation. This has a specific Riemannian metric, under which its Gaussian curvature is not constant and this gives the Klein quartic a Riemannian metric of constant curvature −1 that it inherits from H2. This group is known as PSL, and also as the isomorphic group PSL. By covering space theory, the group G mentioned above is isomorphic to the group of the compact surface of genus 3. It is important to two different forms of the quartic. The closed quartic is what is meant in geometry, topologically it has genus 3 and is a compact space. The open or punctured quartic is of interest in theory, topologically it is a genus 3 surface with 24 punctures. The open quartic may be obtained from the closed quartic by puncturing at the 24 centers of the tiling by regular heptagons, as discussed below. The Klein quartic can be viewed as an algebraic curve over the complex numbers C, defined by the following quartic equation in homogeneous coordinates on P2. The locus of this equation in P2 is the original Riemannian surface that Klein described, note the identity 3 =72, exhibiting 2 - η as a prime factor of 7 in the ring of integers. The group Γ is a subgroup of the triangle group. Namely, Γ is a subgroup of the group of elements of unit norm in the algebra generated as an associative algebra by the generators i, j. One chooses a suitable Hurwitz quaternion order Q H u r in the quaternion algebra, Γ is then the group of norm 1 elements in 1 + I Q H u r. The least absolute value of a trace of an element in Γ is η2 +3 η +2, corresponding the value 3.936 for the systole of the Klein quartic. The Klein quartic admits tilings connected with the group. This tiling is a quotient of the order-3 bisected heptagonal tiling of the hyperbolic plane and this tiling is uniform but not regular, and often regular tilings are used instead