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Topic: Riemann surface

	PlanetMath: Riemann surface
		The simplest example of a Riemann surface which is not a subset of the complex plane is the Riemann sphere.
		A Riemann surface in the narrower sense is a branched covering of the complex plane.
		When one has constructed this surface and convinced oneself that it has the topology of a torus, one is well on one's way to developing an intuitive understanding of Riemann surfaces.
	planetmath.org /encyclopedia/RiemannSurface.html (1824 words)

	Bernhard Riemann Summary (Site not responding. Last check: 2007-10-20)
		Riemann wove together and generalized three crucial discoveries of the 19th century: the extension of Euclidean geometry to n dimensions; the logical consistency of geometries that are not Euclidean; and the intrinsic geometry of a surface, in terms of its metric and curvature in the neighborhood of a point.
		Riemann was born on September 17, 1826 at Breselenz, Hanover, Germany, the son of a Lutheran minister.
		Riemann was shy and self-effacing and recognition for his work came slowly during his lifetime; awareness of his truly striking achievements was to come later as his work was validated and as it stimulated the work of others.
	www.bookrags.com /Bernhard_Riemann (4836 words)

	Riemann surface - Wikipedia, the free encyclopedia
		The Riemann surfaces with curvature 0 are called parabolic; C and the 2-torus are typical parabolic Riemann surfaces.
		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.
		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.
	en.wikipedia.org /wiki/Riemann_surface (1285 words)

	Riemann biography
		Riemann moved from Göttingen to Berlin University in the spring of 1847 to study under Steiner, Jacobi, Dirichlet and Eisenstein.
		In 1859 Dirichlet died and Riemann was appointed to the chair of mathematics at Göttingen on 30 July.
		Riemann considered a very different question to the one Euler had considered, for he looked at the zeta function as a complex function rather than a real one.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Riemann.html (2799 words)

	[No title]
		Riemann first presented to the world his new idea in his doctoral dissertation of 1851, and elaborated its implications in his 1854 habilitation lecture, his 1857 treatises on Abelian and hypergeometric functions, and his posthumously published philosophical fragments.
		Following Gauss, Riemann recognized that in the type of least action physical manifold exemplified by the catenoid or Gauss's potential surfaces, the curves of maximum and minimum curvature are harmonically related, which means that their mutual curvatures change at the same rate, in perpendicular directions.
		Riemann showed, than Abel's extended class of higher transcendental functions, when expressed on Riemann's surface, express a type of transformation that increases the rate and the density at which singularities can be added.
	www.wlym.com /antidummies/part60.html (3651 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-20)
		Riemann (1851–1857, see [1]) was the first to show how for any algebraic function one can construct a surface on which this function can be considered as a single-valued rational function of the point.
		A Riemann surface (with boundary) is a triangulable and orientable manifold with a countable base and, hence, it is separable and metrizable.
		A compact Riemann surface (without boundary) is called a closed Riemann surface; the wider class of finite Riemann surfaces includes the closed Riemann surfaces and the compact Riemann surfaces with a boundary consisting of a finite number of connected components.
	eom.springer.de /r/r082040.htm (2400 words)

	Higher Dimensional Geometry
		Riemann examined surfaces of "positive curvature," like the surface of a sphere, where the angles of a triangle exceed 180°, parallel lines (defined as being great circles of the sphere, not latitudinal lines) always meet, and the shortest distance between two points goes through the surface.
		Riemann's version of wormholes consisted of two flat surfaces connected by a cut (now called a Riemann cut) that is topologically equivalent to the ordinary picture of a wormhole, but lacks a neck.
		Although Riemann himself saw these cuts as objects of geometrical interest and not as methods of traveling between universes or areas, the concept of wormholes as shortcuts through time and space is a popular one within the modern physics community and even today's popular culture.
	library.thinkquest.org /27930/geometry.htm (3058 words)

	Riemann mapping theorem Summary (Site not responding. Last check: 2007-10-20)
		Riemann's proof depended on the Dirichlet principle (whose name was created by Riemann himself), which was considered sound at the time.
		His proof used Riemann surfaces and it was simplified by Paul Koebe two years later in a way which did not require them.
		The Riemann mapping theorem can be generalized to the context of Riemann surfaces: If U is a simply-connected open subset of a Riemann surface, then U is biholomorphic to one of the following: the Riemann sphere, the complex plane or the open unit disk.
	www.bookrags.com /Riemann_mapping_theorem (1845 words)

	BERNHARD RIEMANN
		Riemann was born in 1826 in the kingdom of Hannover, later part of Germany.
		Riemann's essay made considerable progress on this problem, first by giving a criterion for a function to be integrable (or as we now say, Riemann integrable), and then by obtaining a necessary condition for a Riemann integrable function to be representable by a Fourier series.
		Riemann's lecture, "On the hypotheses that lie at the foundation of geometry" was given on June 10, 1854.
	www.usna.edu /Users/math/meh/riemann.html (1057 words)

	Riemann
		Riemann was always very close to his family and he would never have changed courses without his father's permission.
		Gauss did lecture to Riemann but he was only giving elementary courses and there is no evidence that at this time he recognised Riemann's genius.
		Klein, however, was fascinated by Riemann's geometric approach and he wrote a book in 1892 giving his version of Riemann's work yet written very much in the spirit of Riemann.
	www.physics.miami.edu /~curtright/Riemann.html (2622 words)

	Riemann
		However, Riemann's thesis is a strikingly original piece of work which examined geometric properties of analytic functions, conformal mappings and the connectivity of surfaces.
		Riemann's lecture Über die Hypothesen welche der Geometrie zu Grunde liegen (On the hypotheses that lie at the foundations of geometry), delivered on 10 June 1854, became a classic of mathematics.
		In the mathematical apparatus developed from Riemann's address, Einstein found the frame to fit his physical ideas, his cosmology, and cosmogony: and the spirit of Riemann's address was just what physics needed: the metric structure determined by data.
	www.meta-religion.com /Mathematics/Biography/riemann.htm (2626 words)

	Riemann's Minimal Surface (Site not responding. Last check: 2007-10-20)
		The surface is foliated horizontally by circles which become straught lines at the heights of the planar ends.
		One remarkable property of the surface is that it has planar ends at which the Gauß map has degree two.
		The surface comes in fact in a family, it exists on all rectangular tori.
	www.indiana.edu /~minimal/essays/riemann/index.html (112 words)

	Algebraic Curves and Riemann Surface (Site not responding. Last check: 2007-10-20)
		Riemann Surfaces were introduced in the 19th century to help resolve strange formulae and explain mysterious behaviour arising from problems in function theory.
		Riemann's initial discription was confusing to many people and the attempt to understand these objects and his wonderful new ideas led to the growth or creation of many fields of mathematics: topology, the theory of manifolds, potential theory, and algebra to name a few.
		The idea of a Riemann surface is now a central one in mathematics, and appears in such seemingly diverse areas as integrable systems, number theory, algebraic geometry, and string theory.
	www.mast.queensu.ca /~mikeroth/surfaces/surfaces.html (1044 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-20)
		is a harmonic differential on the Riemann surface.
		Of fundamental importance in the theory of differentials on Riemann surfaces is the problem of the existence of a harmonic and analytic differential with given singularities on an arbitrary Riemann surface
		This problem is directly connected with the global uniformization problem for Riemann surfaces, since the construction of a global uniformizing parameter requires the ability to construct a differential with given singularities.
	eom.springer.de /D/d032240.htm (981 words)

	Bernhard Riemann - Wikipedia, the free encyclopedia
		The theory of Riemann surfaces was elaborated by Felix Klein and particularly Adolf Hurwitz.
		Riemann was born in Breselenz on November 17, 1826, a village near Dannenberg in the Kingdom of Hanover in what is today Germany.
		Riemann found that in four spatial dimensions, one needs a collection of ten numbers at each point to describe the properties of a manifold, no matter how distorted it is. This is the famous metric tensor.
	en.wikipedia.org /wiki/Bernhard_Riemann (1023 words)

	Riemann-Surface
		Riemann surfaces were first studied by Bernhard Riemann in his Inauguraldissertation at Göttingen in 1851.
		The result of the construction is a Riemann surface whose points are in one-to-one correspondence with the points of the z-plane.
		The Riemann surface is orientable, since every orientation of a sheet is carried over to the sheet next to it.
	www.eg-models.de /models/Surfaces/Riemann_Surfaces/2002.05.001/_preview.html (393 words)

	18.04 Pictures Home Page
		Riemann Surface associated with a multiple valued function on the complex plane.
		The reason is as follows: when joining the two cut planes to make the surface, we have to join the lower lip of the cut in one plane with the upper in the other (and conversely).
		Riemann Surface is an object in three dimensional space.
	www-math.mit.edu /18.04/18.04-rrr/Pictures/index.html (2269 words)

	Surface Gallery (Site not responding. Last check: 2007-10-20)
		About Riemann's Minimal Surfaces H. Karcher This is the family of singly-periodic embedded minimal surfaces found by Riemann.
		The Gauss map is the Weierstrass pe function additively normalized to have a double zero at the branch point diagonally opposite the double pole and multiplicatively normalized to have the values plus or minus i at the four midpoints (on the Torus) between the zero and the pole.
		The surface is parametrized by the range of the Gauss map with polar coordinates around the punctures.
	xahlee.org /surface/riemann/riemann.html (174 words)

	Riemann Surface - Talks
		Geometric structures are natural structures of surfaces, which enable different geometries to be defined on the surfaces.
		We study the affine structure of domain manifolds in depth and prove that the existence of manifold splines is equivalent to the existence of a manifold's affine atlas.
		Surface geometry is often modeled with irregular triangle meshes.
	www.cise.ufl.edu /~gu/talks/index.htm (884 words)

	NeverEndingBooks » Archive » the noncommutative manifold of a Riemann surface - noncommutative.org
		Here, the checkerboard-surface is part of the Riemann surface and the extra structure consists in putting in each point of the Riemann surface a sphere, reflecting the local structure of the Riemann surface near the point.
		Thus, commutative algebraic geometry of smooth curves (that is Riemann surfaces if you look at the ‘real’ picture) can be seen as the study of one-dimensional representations of their smooth coordinate algebras.
		In other words, the classical Riemann surface gives us already the classifcation of all one-dimensional representations, so nw we are after the ‘other ones’.
	www.neverendingbooks.org /?p=306 (1308 words)

	Math 428: Topics in Complex Analysis, Fall 98
		In more recent terminology, a Riemann surface is understood to be a 1-dimensional complex manifold or equivalently a real 2 dimensional manifold with a distinguished collection of coordinate charts whose transition functions preserve both angles and orientation.
		It is well-known that Riemann surface theory is not only a point of departure but even at the historical roots of a large part of mathematics, in particular, topology, differential geometry, algebraic geometry, algebraic number theory, and partial differential equations.
		Show that S has the structure of a Riemann surface so that the projection map taking a complex number z to its # equivalence class is holomorphic.
	math.rice.edu /~hardt/428 (1647 words)

	Amazon.com: Lectures on Vector Bundles over Riemann Surfaces. (MN-6): Books: Robert C. Gunning,R. C. Gunning (Site not responding. Last check: 2007-10-20)
		Given a sheaf over one Riemann surface M, an analytic mapping from M to another Riemann surface induces a sheaf on the other, and the author studies these induced sheaves in chapter 3.
		By considering for a Riemann surface the mapping which associates to a flat vector bundle its characteristic representation, the author studies families of flat vector bundles in chapter 9.
		Here 'irreducible' is an algebraic geometry notion, and refers to the fact that the homomorphisms from the fundamental group of the Riemann surface to the complex general linear group has the structure of a complex analytic variety.
	www.amazon.com /Lectures-Vector-Bundles-Riemann-Surfaces/dp/0691079986 (1398 words)

	triangle2.nb (Site not responding. Last check: 2007-10-20)
		As an application, we construct a Riemann surface of genus 2 which is tessellated in two different ways by 32 (15°,45°,90°)-triangles.
		It is easy to see that the vertices are glued together in quadruples so that the result is a Riemann surface with a hyperbolic metric.
		Exercise: There is an autorphism of the surface which maps the first to the second tessellation.It has two fixed points so that the quotient of the surface by the order 2 group genrated by this automorphism is a torus.
	www.math.uni-bonn.de /people/weber/mathematica/hyperbolic/Links/triangle_lnk_3.html (184 words)

	Brain Surface Parameterization using Riemann Surface Structure (Site not responding. Last check: 2007-10-20)
		Rather than evolve the surface geometry to a plane or sphere, we instead use the fact that all orientable surfaces are Riemann surfaces and admit conformal structures, which induce special curvilinear coordinate systems on the surfaces.
		The resulting surface subdivision and the parameterizations of the components are intrinsic and stable.
		To illustrate the technique, we computed conformal structures for several types of anatomical surfaces in MRI scans of the brain, including the cortex, hippocampus, and lateral ventricles.
	www.math.ucla.edu /~ylwang/doc/riemann.html (329 words)

	Riemann Surface - Tutorial (Site not responding. Last check: 2007-10-20)
		The images, movies and surface Ricci flow part are prepared by David Gu and the students, Miao Jin, Hongyu Wang and Xiaotian Yin.
		It covers the major concepts in topology, differential geometry, Riemann surface theories related to conformal geometry, and many major algorithms for computing conformal structures.
		It includes some examples for computing flat metrics on surfaces with a single cone singularity, in other words, all curvature is concentred on a single point.
	www.cise.ufl.edu /~gu/tutorial/index.htm (279 words)

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