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Hyperbolic Space-Time, and Hyperbolic Numerical Manifolds As a subset of a hyperbolic manifold, the numerical logic of space-time likewise is hyperbolic. The superior manifold is characterized by the Euler function, or e-hyperbola. It is the hyperbolic nature of both the superior numerical manifold, and the lesser manifold subtending space-time, which is responsible for the phenomenon of dimensionality. members.aol.com /YuzhnoeMore/arg1.html   (1141 words)

[No title] Given a vector space of functions of a parameter or functions on a manifold, an operator may have a kernel or matrix whose rows and columns are indexed by the parameter or by points on the manifold. Orbifolds are manifolds with singularities such as reflection surfaces, where they resemble manifolds with boundary, and cone lines, where they are modelled (in the direction perpendicular to the cone line) by a cone with an angle of 360/n degrees for some n. PL flow A "piecewise linear" motion on a space or a manifold, akin to a flow given by a vector field, in which every particle in a given simplex of some triangulation moves with constant velocity and in the same direction, so that the particle trajectories are polygons. www.ornl.gov /sci/ortep/topology/defs.txt   (5717 words)

PlanetMath: stable manifold This result is also valid for nonperiodic points, as long as they lie in some hyperbolic set (stable manifold theorem for hyperbolic sets). Cross-references: hyperbolic set, unstable, tangent spaces, stable manifold theorem, hyperbolic periodic point, diffeomorphism, smooth manifold, compact, subset, invariant, metric, point, metrizable, neighborhood, least period, periodic point, fixed point, homeomorphism, topological space This is version 8 of stable manifold, born on 2003-06-13, modified 2006-06-28. planetmath.org /encyclopedia/StableManifold.html   (196 words)

UC Davis Math: Glossary   (Site not responding. Last check: 2007-10-10) A manifold with the property that each tangent space has the structure of a complex vector space, but the complex structures are not necessarily compatible with true complex coordinates as they are for a complex manifold. A vertex at infinity of a hyperbolic polyhedron. A motion on a space or a manifold, akin to a flow given by a vector field, in which every particle in a given simplex of some triangulation moves with constant velocity and in the same direction, so that the particle trajectories are polygons. www.math.ucdavis.edu /profiles/glossary.html   (9932 words)

PlanetMath: hyperbolic fixed point If the dimension of the stable manifold of a fixed point is zero, the point is called a source; if the dimension of its unstable manifold is zero, it is called a sink; and if both the stable and unstable manifold have nonzero dimension, it is called a saddle. Cross-references: stable, point, stable manifold, dimension, iterate, least period, periodic point, linear hyperbolic isomorphism, diffeomorphism, fixed point, smooth manifold This is version 3 of hyperbolic fixed point, born on 2003-07-27, modified 2003-07-29. planetmath.org /encyclopedia/HyperbolicFixedPoint.html   (149 words)

Hyperbolic 3-manifold - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab2.tamu.edu)   (Site not responding. Last check: 2007-10-10) A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant sectional curvature -1. In other words, it is the quotient of three-dimensional hyperbolic space by a subgroup of hyperbolic isometries acting freely and properly discontinuously. A cusped hyperbolic 3-manifold is a hyperbolic 3-manifold with at least one cusp. en.wikipedia.org.cob-web.org:8888 /wiki/Hyperbolic_3-manifold   (331 words)

The Computing Logo   (Site not responding. Last check: 2007-10-10) The computing logo is part of the the horoball packing for a hyperbolic manifold with two cusps, v3551 in Jeff Weeks's census of orientable cusped hyperbolic manifolds with seven tetrahedra. It is a standard spine of v3551, and by deletin g the large 24 sided two-cell we obtain a one sided, nonorientable surface of Euler characteristic -1 embedded in v3551. The cousins of v3551 are the census manifolds v3501, v3530, v3537, v3547, v3549, v3550, and v3551. www.msri.org /local/computing/logo   (372 words)

Petra Bonfert-Taylor's publications (via CobWeb/3.1 planetlab2.tamu.edu)   (Site not responding. Last check: 2007-10-10) We show that any closed hyperbolic surface admitting a conformal automorphism with ``many'' fixed points is uniformly quasiconformally homogeneous, with constant uniformly bounded away from 1. In dimension great than or equal to 3, a hyperbolic manifold is uniformly quasiconformally homogeneous if and only if it is a regular cover of a closed hyperbolic orbifold. We show that a discrete, quasiconformal group preserving n-dimensional hyperbolic space has the property that its exponent of convergence and the Hausdorff dimension of its limit set detect the existence of a non-empty regular set on the sphere at infinity. pbonfert.web.wesleyan.edu.cob-web.org:8888 /publist.html   (1204 words)

Andrew Przeworski's Homepage   (Site not responding. Last check: 2007-10-10) A universal upper bound on density of tube packings in hyperbolic space (dvi+ps, Postscript, PDF file): Hyperbolizes a Euclidean result to produce an upper bound on density of tube packings in hyperbolic space. Volumes of hyperbolic 3-manifolds of betti number at least 3 (dvi, Postscript, PDF): Shows that any closed orientable hyperbolic 3-manifold with betti number at least 3 has volume at least 1.015. Cones embedded in hyperbolic manifolds (dvi with eps, Postscript, PDF, Journal version): Establishes a lower bound on the volume outside of a maximal tube. www.math.okstate.edu /~aprzewo   (372 words)

Atlas: Compact hyperbolic 4-manifolds of small volume by Marston Conder   (Site not responding. Last check: 2007-10-10) In the case of dimension 4 (as with all even dimensions), the volume of a hyperbolic manifold is a constant multiple of its Euler characteristic. Ratcliffe and Tschantz (2000) proved the existence of non-compact orientable hyperbolic 4-manifolds of minimal Euler characteristic 1, and until recently, the compact orientable hyperbolic 4-manifold of smallest known volume was the Davis manifold (1985), which has Euler characteristic 26. Furthermore, these two 4-manifolds and the Davis manifold are all arithmetic, and have the same arithmetic structure, and hence are commensurable (in that they have a common finite cover). atlas-conferences.com /cgi-bin/abstract/caog-16   (194 words)

Body   (Site not responding. Last check: 2007-10-10) Draw a picture similar to Figures 20.2 and 20.3, for the half turn manifold, which is the same as the quarter turn manifold except that it is obtained by gluing the top and bottom faces with a half turn. Five of the six orientable Euclidean manifolds are the 3-torus, the quarter turn manifold, the half turn manifold, the one-sixth turn manifold, and the one-third turn manifold. We saw in Problem 11.6 that the area of a spherical or hyperbolic 2-manifold is determined by its topology and the radius (or curvature) of the model. www.math.cornell.edu /~dwh/books/eg99/Ch20/Ch20.html   (4024 words)

Hyperbolic space - Wikipedia, the free encyclopedia Hyperbolic space is the principal example of a space exhibiting hyperbolic geometry. Hyperbolic spaces may be regarded as models, over the real numbers, of hyperbolic geometry, that is, they satisfy the axioms of hyperbolic geometry. As a result, the universal cover of any closed manifold M of constant negative curvature −1, which is to say, a hyperbolic manifold, is H en.wikipedia.org /wiki/Hyperbolic_space   (625 words)

Warwick Mathematics Institute Hyperbolic Geometry Seminars We investigate the relationship between $\sumb_n$ and $\mathbf{K}(1b_n)$ with hyperbolic geometry and use this geometry to construct a sequence $b_n$ of real numbers for which both $\sum b_n$ and $\mathbf{K}(1b_n)$ converge, thereby answering Wall's question. In this case the pair of binding laminations is replaced by a doubly incompressible lamination on the boundary of the manifold. A key tool for obtaining this decomposition is the so-called "convex hull construction" of a cell decomposition of a hyperbolic manifold, which will also be explained in the talk. www.maths.warwick.ac.uk /~jaram/seminar0405.html   (1456 words)

Edward Taylor's Publications   (Site not responding. Last check: 2007-10-10) Our main result is that the marked, geometrically finite (infinite volume) hyperbolic 3-manifolds homotopy equivalent to M are open in the topology defined by strong convergence on the space of all marked hyperbolic 3-manifolds homotopy equivalent to M. This in turn implies that the two hyperbolic 3-manifolds are themselves K-quasi-isometric, where the distortion K decreases as the operator norm of the difference of the scattering operators decreases. In particular, we show that for a pair of geometrically isomorphic convex co-compact Kleinian groups, the ratio of the mass of the Patterson-Sullivan measure on line element space to the mass of its push-forward is bounded below by the ratio of the Hausdorff dimensions of the limit sets. ectaylor.web.wesleyan.edu /publist.html   (1452 words)

[No title] For example, Waldhausen proved in the 1960's that a Haken manifold M is determined up to homeomorphism by its fundamental group (`algebra determines topology'). For example, since Gabai and Oertel proved that a laminar manifold has universal cover $\bf R^3$, Delman and Roberts' result immediately proves both the Property P Conjecture (non-trivial Dehn surgery never yields a simply-connected manifold) and the Cabling Conjecture (Dehn surgery on a non-cabled knot cannot yield a reducible manifold) for alternating knots. However, current sentiment favors the existence of a non-laminar, hyperbolic 3-manifold; some of the best prospects, in fact, seem to be among some of the other surgeries on the (-2,3,7) pretzel knot. www.math.unl.edu /~mbrittenham2/personal/myresold.html   (1253 words)

Springer Online Reference Works Kobayashi hyperbolicity describes in a precise sense whether a complex manifold contains arbitrarily large copies of a one-dimensional complex disc. The former is an example of a Kobayashi-hyperbolic manifold while the latter has arbitrarily large discs in it and is not Kobayashi hyperbolic. Green, see [a2], who gave some criteria ensuring that the complement of a finite family of complex hypersurfaces in complex projective space is Kobayashi hyperbolic. eom.springer.de /k/k110120.htm   (338 words)

Asymptotic Group Theory (via CobWeb/3.1 planetlab2.tamu.edu)   (Site not responding. Last check: 2007-10-10) Abstract: The isometry group of a closed hyperbolic n-manifold is finite. We prove that for every n>1 and every finite group G there is an n-dimensional closed hyperbolic manifold whose isometry group is G. This resolves a longstanding problem whose low-dimensional case n=2 and n=3 were proved by Greenberg ('74) and Kojima ('88) respectively. Applications to relatively hyperbolic groups include descriptions of relatively hyperbolic groups with infinite Out(G) and of co-Hopfian relatively hyperbolic groups. www.sci.ccny.cuny.edu.cob-web.org:8888 /~shpil/abstracts.html   (366 words)

Chapter 5 - Rotations of the Binary Tree and the Modular Group   (Site not responding. Last check: 2007-10-10) Note that by imposing the modular group symmetry on the real number line, we've essentially introduced a hyperbolic manifold that is homomorphic to the real-number line. Of course they do, since their 'true' trajectories should be considered to live on the hyperbolic manifold rather than on the real-number line. Higher dimensional manifolds seem to be generated by quaternions and the octonians. linas.org /math/rotations/rotations.html   (1051 words)

3-manifold - Wikipedia, the free encyclopedia In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is usually made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds. One can choose the surface to be nicely placed in the 3-manifold, which leads to the idea of an incompressible surface and the theory of Haken manifolds, or one can choose the complementary pieces to be as nice as possible, leading to structures such as Heegaard splittings, which are useful even in the non-Haken case. en.wikipedia.org /wiki/3-manifold   (480 words)

Hyperbolic manifold - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab2.tamu.edu)   (Site not responding. Last check: 2007-10-10) In mathematics, a hyperbolic n-manifold is a complete Riemannian n-manifold of constant sectional curvature -1. Its thick-thin decomposition has a thin part consisting of tubular neighborhoods of closed geodesics and/or ends which are the product of a Euclidean n-1-manifold and the closed half-ray. The hyperbolic structure on a hyperbolic n-manifold is unique by Mostow rigidity and so geometric invariants are in fact topological invariants. en.wikipedia.org.cob-web.org:8888 /wiki/Hyperbolic_manifold   (134 words)

BAM: Manifold Destiny, The Classes, May 1997   (Site not responding. Last check: 2007-10-10) I'd read the theorem: "Any closed irreducible three-manifold which is homotopy equivalent to a closed hyperbolic three-manifold is indeed a hyperbolic three-manifold." I was hoping he would have pictures. "That's a one-dimensional manifold," he says, pointing to the circle, "and the arc is a piece of a one-manifold." He draws a sphere, then a doughnut (a torus), then two doughnuts linked as if the baker had forgotten to shut off the dough machine, then chains of three and four more doughnuts. The result, billions of computations later, was the Rigidity Theorem, which the geometry world hailed as a major advance toward proving that many three-manifolds have natural geometric structures. www.brownalumnimagazine.com /storydetail.cfm?ID=1074   (429 words)

The visual core of a hyperbolic $3$-manifold   (Site not responding. Last check: 2007-10-10) In this note we introduce the notion of the visual core of a hyperbolic 3-manifold N = H Applying the results from an earlier paper, we are able to conclude that if the algebraic limit of a sequence of isomorphic Kleinian groups is a generalized web group, then the visual core of the algebraic limit manifold embeds in the geometric limit manifold. However, we show that the (interior of the) visual core of a summand does embed in the resulting manifold. www.math.lsa.umich.edu /~canary/visualab.html   (306 words)

Real Projective Structures on Hyperbolic Manifolds There is a model of the hyperbolic plane known as the Klein Model in which points in the hyperbolic plane are thought of as points on the interior of a conic in The right hexagons are determined by the lengths of every other side, and so the hyperbolic structure on a 3-punctured sphere is determined by the lengths of the geodessics around the punctures. After constructing our surface we can change its hyperbolic structure later by cutting along one of our geodessics, and reattaching the (now) two circles in a different way (intuitively we will rotate one of the circles dragging part of the surface along with it and regluing). merganser.math.gvsu.edu /david/reed03/projects/hooper   (1134 words)

Quasars and Redshift   (Site not responding. Last check: 2007-10-10) Before 1998 it was thought that a universe that expanded forever would have a hyperbolic ("open") manifold, and a universe which would eventually collapse would have a spherical ("closed") manifold. Since 1998 the standard model has become that of an open forever-expanding universe, but the old idea that this would entail a hyperbolic manifold has not come with it. It's not only possible, but mandatory that a hyperbolic space be enclosed by a spherical, as hyperbolic space carries with it an asymptote as a boundary point, i.e. quasars.org   (1079 words)

On the -cohomology of a convex cocompact hyperbolic manifold, Xiaodong Wang On the -cohomology of a convex cocompact hyperbolic manifold, Xiaodong Wang We prove a vanishing theorem for a convex cocompact hyperbolic manifold which relates its L The borderline case is shown to characterize the manifold completely. projecteuclid.org /getRecord?id=euclid.dmj/1085598144   (269 words)

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