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# Topic: Riemann sphere

###### In the News (Thu 22 Aug 19)

 Riemann surface - Wikipedia, the free encyclopedia 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. 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. For every closed parabolic Riemann surface, the fundamental group is isomorphic to a rank 2 lattice group, and thus the surface can be constructed as C/Γ, where C is the complex plane and Γ is the lattice group. en.wikipedia.org /wiki/Riemann_surface   (1193 words)

 Sphere - Wikipedia, the free encyclopedia The sphere has the smallest surface area among all surfaces enclosing a given volume and it encloses the largest volume among all closed surfaces with a given surface area. For this reason, the sphere appears in nature: for instance bubbles and small water drops are roughly spherical, because the surface tension minimizes surface area. The circumscribed cylinder for a given sphere has a volume which is 3/2 times the volume of the sphere, and also a surface area which is 3/2 times the surface area of the sphere. en.wikipedia.org /wiki/Sphere   (883 words)

 Sphere   (Site not responding. Last check: 2007-11-06) Sphere is the name of a book written by Michael Crichton, which was subsequently turned into a movie by the same name. More precisely, a sphere is the set of points in 3-dimensional Euclidean space which are at distance r from a fixed point of that space, where r is a positive real number called the radius of the sphere. For this reason, the sphere appears in nature: for instance bubbles and small water drops are spheres, because the surface tension tries to minimize surface area. www.sciencedaily.com /encyclopedia/sphere   (633 words)

 Riemann sphere - Wikipedia, the free encyclopedia In the category of Riemann surfaces, the automorphism group of the Riemann sphere is the group of Möbius transformations. The Riemann sphere is one of three simply-connected Riemann surfaces. This statement, known as the uniformization theorem, is important to the classification of Riemann surfaces. en.wikipedia.org /wiki/Riemann_sphere   (351 words)

 Riemann surface: Definition and Links by Encyclopedian.com - All about Riemann surface   (Site not responding. Last check: 2007-11-06) In complex analysis, a Riemann surface is a one-dimensional complex manifold. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially "multi-valued" ones such as the square root or the logarithm. Every simply connected Riemann surface is conformally equivalent to C or to the Riemann sphere C ∪ {∞} or to the open disk {z ∈ C : www.encyclopedian.com /ri/Riemann-sphere.html   (1371 words)

 eLibrary Project : 3-sphere   (Site not responding. Last check: 2007-11-06) In mathematics, a "3-sphere" is a higher-dimensional analogue of a sphere. A regular sphere, or 2-sphere, consists of all points equidistant from a single point in ordinary 3-dimensional Euclidean space, "R" A 3-sphere consists of all points equidistant from a single point in "R" Whereas a 2-sphere is a smooth surface,2-dimensional surface, a 3-sphere is an object with three dimensions, also known as 3-manifold. Roughly speaking, a glome is to a sphere as a sphere is to a circle. elibraryproject.com /info/3-sphere.html   (676 words)

 Sphere   (Site not responding. Last check: 2007-11-06) In non-mathematical usage, the term sphere is often used for something "solid" (which mathematicians call ball). But in mathematics, sphere refers to the boundary of a ball, which is "hollow". The sphere is the inverse image of a one-point set under the continuous function ''x''. sphere.search.ipupdater.com   (685 words)

 Riemann sphere -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-11-06) The Riemann sphere is named after the geometer (Pioneer of non-Euclidean geometry (1826-1866)) Bernhard Riemann. When the sphere is given the round (A system of related measures that facilitates the quantification of some particular characteristic) metric the (Click link for more info and facts about isometry group) isometry group is the subgroup PSU The Riemann sphere is one of three (Click link for more info and facts about simply-connected) simply-connected Riemann surfaces. www.absoluteastronomy.com /encyclopedia/R/Ri/Riemann_sphere.htm   (618 words)

 [No title] As Riemann emphasized in his 1854 habilitation lecture, his philosophical fragments, and his lectures on Abelian and hypergeometric functions, and as Gauss had spoken earlier, the common, intuitive notions concerning geometrical objects presupposes a set of assumptions concerning the fundamental nature of the "space" in which those objects arise. On this basis, Riemann was able to demonstrate, using Leibniz's principle of "analysis situs", that all the essential characteristics of a complex function, are determined by the relationship of the boundary to the branching points, or the other singularities. Riemann demonstrated, using a method similar to the one used by Gauss in his proof of the fundamental theorem of algebra, that those complex functions associated with these higher transcendentals generate surfaces with two sheets, but with more than the two branch points expressed by the algebraic, circular, hyperbolic or exponential functions. www.wlym.com /antidummies/part56.html   (4637 words)

 Riemann sphere Definition / Riemann sphere Research   (Site not responding. Last check: 2007-11-06) [click for more], the Riemann sphere is the unique simply-connected A geometrical object is named simply connected if it consists of one piece and doesn't have any circle-shaped "holes" or "handles". Riemann sphere is the unique simply-connected, compact, Riemann surface. Riemann sphere is a method of visualizing the complex plane union the point at infinity (commonly referred to as positive infinity). www.elresearch.com /Riemann_sphere   (248 words)

 Sphere   (Site not responding. Last check: 2007-11-06) A fused quartz gyroscope for the Gravity Probe B experiment which differs from a perfect sphere by no more than a mere 40 atoms of thickness as it refractionrefracts the image of Einstein in the background. The circumscribed Cylinder (geometry)cylinder for a given sphere has a volume which is 3/2 times the volume of the sphere, and also a surface area which is 3/2 times the surface area of the sphere. The memorial consists of a concrete spiral with a sphere in the center. www.infothis.com /find/Sphere   (952 words)

 [No title]   (Site not responding. Last check: 2007-11-06) A Riemann surface is a topological space with a collection of "charts": homeomorphisms phi_i of open subsets of X to open subsets of the complex plane, such that the charts cover X, and wherever two charts overlap, the functions phi_i composed with {phi_j}^{-1} is a holomorphic function. By the Riemann mapping theorem, though, there is an isomorphism between any two bounded simply connected open subsets of the complex plane; there's a 1-1 map from one to the other which has an inverse which is also holomorphic. You think of a sphere sitting on the complex plane at the origin, and each point is identified with the point on the sphere which is collinear with the given point and the top of the sphere. www.math.niu.edu /~rusin/papers/known-math/95/modularity   (2492 words)

 Complex Calculator In the starting position of the sphere (press "Reset" to achieve it in case you have rotated the sphere), 0 is the South pole of the sphere, infinity the North pole, and the equator represents the circle of radius 1 around 0. In cartography, the world sphere is mapped onto a sheet of paper by the reverse process of the way the plane is mapped onto the Riemann sphere. The Riemann sphere is always useful to observe a neighbourhood of infinity, but there are some functions that are special in relation to the sphere. www.dcs.gla.ac.uk /~bunkenba/CC/CC.html   (2286 words)

 Riemann Sphere   (Site not responding. Last check: 2007-11-06) The use of a unit sphere is merely to simplify both the notation and computation. The sphere need not be of a unit radius however for this mapping to be consistent. The goal is to define the mapping from the point (0,0,1) on the north pole through the point (a,b,c) on the surface of the sphere onto the complex plane passing through the equator of the sphere. www.mindspring.com /~thumper5/chadpg/riemann.html   (339 words)

 Animations in Complex Dynamics   (Site not responding. Last check: 2007-11-06) The Julia set divides the Riemann sphere into the basin of attraction of 0, the Baker domain, and all preimages of the basin and of the Baker domain. In this animation, the Julia set is shown in white on the Riemann sphere. All pictures are made on the Riemann sphere in a chart near infinity. www.math.uiuc.edu /~aimo/anim.html   (401 words)

 Structural stability in holomorphic dynamics (Spring 2003)   (Site not responding. Last check: 2007-11-06) The study of iterations of rational mappings of the Riemann sphere was started at the end of 19-th - beginning of 20th century, in the works of Fatou and Julia. The Teichmueller distance between Riemann surfaces induces a Finsler metric on this manifold. This yields the classification of the automorphisms of the space of Riemann surfaces. www.mccme.ru /ium/s03/stabholdyn.html   (376 words)

 Physics Help and Math Help - Physics Forums - Riemann's Translucent ball If take the radius of the sphere to be R and let andphi; be the angle the line through the center of the sphere and the point x on the x-axis, then andphi;= tan The Riemann sphere is talking about a geometric representation of numbers- it has no relationship whatsoever with a decimal representation. If we reduce Riemann's ball to a single circle that goas through south pole (= 0) and north pole (= oo) then 1 is the middle point (on the circle's line) between 0 and oo. www.physicsforums.com /printthread.php?t=9199   (760 words)

 Sphere to Sphere   (Site not responding. Last check: 2007-11-06) from the n dimensional sphere to the n dimensional sphere, study the degree of the map. Its not hard to show that the degree of the polynomial is in fact equal to the degree deg(f) when the polynomial is viewed as a continuous function on the Riemann sphere. Below, I have drawn the image on the Riemann sphere of the annulus 0.9complex plane under the action of a complex polynomial. www.math.neu.edu:16080 /~lovett/math/s2tos2.htm   (183 words)

 Coalescence on Riemann Surfaces (ResearchIndex)   (Site not responding. Last check: 2007-11-06) Riemann Surface of genus g = 0) is determined by the operator product expansions to Q i!j (z i \Gamma z j) Q i!j (y j \Gamma y i) Q N i;j=1 (z i \Gamma y j) \Gamma1. 52.5%: On coalescence of fermions on Riemann surfaces - Matthias Schork Fb On coalescence of fermions on Riemann surfaces - Matthias Schork Fb citeseer.ist.psu.edu /283463.html   (343 words)

 Station Information - Riemann sphere In mathematics, the Riemann sphere of complex analysis is a compact Riemann surface, which consists of a complex plane and a point at infinity. It can be represented by an atlas with two charts, and a transition function z=1/w where w and z are nonzero. The Riemann sphere can be conformally mapped onto a euclidian plane by stereographic projection. www.stationinformation.com /encyclopedia/r/ri/riemann_sphere.html   (67 words)

 Rotations of Riemann Sphere   (Site not responding. Last check: 2007-11-06) Which linear fractional transformations induce rotations of the sphere? These notes supplement the discussion of linear fractional mappings presented in a beginning graduate course in complex analysis. The goal is to prove that a mapping of the Riemann sphere to itself is a rotation if and only if the corresponding map induced on the plane by stereographic projection is a linear fractional whose (two-by-two) coefficient matrix is unitary. www.mth.msu.edu /~shapiro/Pubvit/Downloads/RS_Rotation/rotation.html   (68 words)

 Slide 1 Möbius transformations are continuous on the Riemann sphere The North pole of the Riemann sphere corresponds to     , the South pole corresponds to the origin of the complex plane and the equator corresponds to the unit circle. Therefore, M(z) is continuous on the extended z plane, or Riemann sphere. www.sju.edu /~rhall/Complex/moebius1_files/slide0003.htm   (82 words)

 The Gauss Plane and the Riemann Sphere   (Site not responding. Last check: 2007-11-06) If one adds a point at infinity to the Gauß plane one obtains a sphere, often called the Riemann sphere. Complex numbers in the plane are associated to points on the sphere via the stereographic projection. Its basic idea is that a non-horizontal line through the north pole of the sphere intersects the sphere in exactly one additional point, and this point is associated to the point of intersection between the line and the Gauß plane. www.ualberta.ca /dept/math/gauss/fcm/Complex/Numbers/RmnnSphr.htm   (159 words)

 [No title] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% McMullen, Curt %% %% { Riemann surfaces and the geometrization of 3-manifolds %% %% publ: Bull. Our argument rests on a result entirely in the theory of Riemann surfaces: an extremal quasiconformal mapping can be relaxed (isotoped to a map of lesser dilatation) when lifted to a sufficiently large covering space (e.g., the universal cover). The theorem states that given any two Riemann surfaces $X, Y \in \Teic(S)$ there is a unique 3-manifold interpolating between them; this is {\em Bers' simultaneous uniformization theorem} \cite{Bers:simunif}. www.ams.org /journals/bull/pre-1996-data/199227-2/McMullen   (3346 words)

 Articles - Lorentz group   (Site not responding. Last check: 2007-11-06) Under stereographic projection from the Riemann sphere to the Euclidean plane, the effect of this MÃ¶bius transformation is a dilation from the origin. The corresponding continuous transformations of the celestial sphere move points along a family of circles which are all tangent at the North pole to a certain great circle. The MÃ¶bius transformations are precisely the conformal transformations of the Riemann sphere (or celestial sphere). www.lastring.com /articles/Lorentz_group   (3246 words)

 [No title]   (Site not responding. Last check: 2007-11-06) Vladimir Lin, Technion TITLE: Holomorphic self-mappings of Riemann sphere's configuration spaces DATE: Monday, December 4, 2000 PLACE: Amado 232 TIME: 15:30 Refreshments will be served in the Faculty Lounge, Room 820, before the Colloquium. ABSTRACT: The configuration space $\mathcal C^n$ of the Riemann sphere $\overline{\mathbb C}$ is the space of all subsets $Y\subset\overline{\mathbb C}$ consisting of $n$ distinct points. The standard action of the M\"obius transformation group $PSL(2,\mathbb C)$ on the Riemann sphere induces the diagonal action of this group in $\mathcal C^n$ defined by $g(\{p_1,...,p_n\})=\{gp_1,...,gp_n\}$ for any subset $\{p_1,...,p_n\}\subset\overline{\mathbb C}$ and any element $g\in PSL(2,\mathbb C)$. www.math.technion.ac.il /~techm/20001204153020001204lin   (143 words)

 Mobius Transformations and The Night Sky This leads to the remarkable fact that the combined effect of any proper orthochronous (and homogeneous) Lorentz transformation on the incidence angles of light rays at a point corresponds precisely to the effect of a particular LFT on the Riemann sphere via ordinary stereographic projection from the extended complex plane. The complex number p in the extended complex plane is identified with the point p' on the unit sphere that is struck by a line from the "North Pole" through p. Relative to an observer located at the center of the Riemann sphere, each point of the sphere lies in a certain direction, and these directions can be identified with the directions of incoming light rays at a point in spacetime. www.mathpages.com /rr/s2-06/2-06.htm   (1843 words)

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