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Topic: Symmetric space

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	Riemannian symmetric space - Wikipedia, the free encyclopedia
		In mathematics, a (Riemannian) symmetric space in differential geometry is a certain kind of homogeneous space in the theory of Lie groups.
		A Lie group characterisation of symmetric spaces is as G/H where G is a Lie group and H a compact subgroup that is open in the fixed set of an automorphism of G of order 2.
		A symmetric space is homogeneous, so can be written as G/H where G is a Lie group acting on it and H is the subgroup fixing some fixed point.
	en.wikipedia.org /wiki/Riemannian_symmetric_space (842 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		A symmetric space, or more general a homogeneous space with an invariant metric, has a transitive Lie group of of isometries.
		And, since in the situation in question being a symmetric space or not means to have a parallel transported curvature tensor or not, it is a cheat not wishing to talk about curvature (as it is, by the way, to try to understand any given Riemannian geometry without talking about curvature.
		In particular, in the case of symmetric spaces geometry IS Lie group theory, in the spirit of Klein.
	www.math.niu.edu /~rusin/papers/known-math/99/symmetric_sp (714 words)

	No Title
		We compute the asymptotics of the number of integer points on affine homogeneous varieties G/H, under the assumption that H is an affine symmetric subgroup of G. We use the Howe-Moore theorem and a certain geometric property of affine symmetric spaces.
		In this paper we investigate when the space of translates of an algebraic measure is relatively compact in the weak-star topology on a locally symmetric space.
		These numbers are equal to the volumes of the moduli spaces of pairs (curve, holomorphic differential) with fixed multiplicities of zeros of the differential and have several applications in ergodic theory.
	www.math.uchicago.edu /~eskin/abstracts.html (1599 words)

	Configuration space geometry
		Configuration space as the space of all possible 3-surfaces in M^4_+\times CP_2 endowed with metric and spinor structure.
		Configuration space geometry must have Kähler property, which means the existence of an antisymmetric tensor giving representation of the imaginary unit in the tangent space of the configuration space.
		The recently found group theoretical construction of the configuration space metric leads to a very explicit form of the metric and the general properties of the metric are the same as those associated with the metric deduced from the proposed Kähler function.
	www.physics.helsinki.fi /~matpitka/cspace.html (524 words)

	The Origin of Gravitation
		The compatibility constraint is surmounted by extracting time directly from space - the flip of the electromagnetic coin from wavelength to frequency - and the closure, entropy, and causality constraints are met by providing the dimension itself, rather than the massive energy form, with an "infinite" one-way velocity which is the metric equivalent of c.
		We can visualize this secondary process as the actual symmetric flow and annihilation of the spatial dimensions, leaving in their place an uncanceled time residue whose intrinsic motion - at right angles to all three spatial dimensions - pulls space after it, producing the continuous spatial collapse that is a gravitational field (see fig.
		As space self-annihilates at the entrance to the time line, a time residue is continuously extracted, which in turn moves on down the time line, pulling more space after it, and so feeding the process and continuing the cycle indefinitely.
	www.people.cornell.edu /pages/jag8/Time.html (5257 words)

	Symmetric space - Wikipedia, the free encyclopedia
		In mathematics, the term symmetric space has several different meanings.
		In general topology, a symmetric space, or R
		space, is a topological space whose Kolmogorov quotient is T
	en.wikipedia.org /wiki/Symmetric_space (95 words)

	Loeks Symmetric Spaces Page
		In order to study p-adic symmetric spaces and their representation theory it is important to have a classification of the k-involutions together with their fine structure of restricted root systems with multiplicities and Weyl groups.
		Motivation: That a study of p-adic symmetric spaces is a natural extension of the study of real semisimple symmetric spaces and also of p-adic groups is not easily seen.
		Recall that the Riemannian symmetric spaces can be defined as those Riemannian manifolds M such that any point x in M has a normal neighborhood on which the geodesic symmetry with respect to x is an isometry.
	www4.ncsu.edu /~loek/research/symm.html (1231 words)

	Gravitation, Entropy, and Thermodynamics
		The flight of space ("wavelength") from time ("frequency") produces the intrinsic motion of light, a symmetric spatial state of energy fleeing an asymmetric temporal state which is nevertheless an internal potential of its own nature (the proverbial "bur under the saddle").
		Space is the only source of temporal entropy, and by consuming space gravity ensures that it is only the entropy account of space and light which is transferred to the entropy account of time and matter.
		That flow is a measure of the energetic difference between the symmetric spatial entropy drive of the free energy which created Earth's mass, the intrinsic motion of light as gauged by "velocity c", and the asymmetric temporal entropy drive of Earth's bound energy, the intrinsic motion of time as gauged by "velocity T".
	www.people.cornell.edu /pages/jag8/thermo.html (9522 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Time: 4:15 PM (Tea at 3:45 PM) Abstract A symmetric operator space is a Banach space of closed operators affiliated with and measurable with respect to a distinguished trace on a given semi-finite von Neumann algebra.
		Important examples of symmetric operator spaces include the familiar (commutative) rearrangement-invariant function spaces as well as the so-called (non-commutative) trace ideals of compact operators on a Hilbert space.
		Symmetric operator spaces form a very wide class of Banach spaces which are, in general, not isomorphic to any of classical Banach spaces.
	www.math.technion.ac.il /~techm/oldmessages/2707 (262 words)

	Event-Symmetric Space-Time (Site not responding. Last check: 2007-10-08)
		This group is known as the symmetric group on the manifold.
		Since the symmetric group acting on space-time can be regarded as a discrete extension of the diffeomorphism group in general relativity, it is worth noting that the diffeomorphism invariance is not all that evident either.
		The analogy is not perfect since it suggests that curved space-time is embedded in some higher dimensional flat space, when in fact, the mathematical formulation of curvature avoids the need for such a thing.
	adela.karlin.mff.cuni.cz /~motl/Gibbs/esst.htm (3265 words)

	A non-symmetric space-time metric (Site not responding. Last check: 2007-10-08)
		The basis of the relationship is that the symmetric part represents the first difference of the vector potential, and the antisymmetric part represents the first difference of the scalar potential.
		In being symmetric in static solutions, the first difference of the vector potential is evidently related to the expansion factor of the gravitational field, which is known to behave similarly to the Newtonian gravitational potential.
		The symmetric terms are investigated on another page.
	www.s-4.com /pulsar/metric.htm (521 words)

	research
		In particular, the basic notions for conformal immersions from a Riemann surface into the 4-sphere and for surfaces in Euclidean space are expressed in terms of the quaternionic calculus, e.g., the mean curvature sphere, the Hopf fields, the Willmore functional, the mean curvature vector, and Gauss and normal curvature.
		Moreover, the deformation h -> h(p,.) of holomorphic curves in complex projective space is linear, i.e., the hyperplane bundle h(p,.) moves linearly in the Jacobian of the spectral curve tangent to its Abel image.
		LPP04] that f comes from the twistor projection of a holomorphic curve in complex projective space or from a minimal surface in 4-space: The associated family of Willmore surfaces is described by an S1-family of flat connections.
	www.gang.umass.edu /~leschke/Public/research/research (3313 words)

	FAH Excerpt: Separation (Site not responding. Last check: 2007-10-08)
		space, and every preregular space is also a symmetric space.
		The two entries in each row of the chart are closely related: a space satisfies the condition in the left column if and only if the space is Kolmogorov and satisfies the condition in the right column in the same row.
		spaces, but the abstract theory can be developed more clearly if we classify properties according to the various axioms in the chart.
	math.vanderbilt.edu /~schectex/ccc/excerpts/separat.html (481 words)

	[No title]
		We discuss the Soule - Lannes method of replacing the symmetric space by a smaller space Z for which the quotients by 0 = SL(3; Z) and the congruence subgroups 1 and 2 are compact.
		The space X2(n) is an (n - 1)-dimensional simplicial complex which can be described as follows: an n - dimensional 2 - adic lattice L is a Z2 - submodule of Qn2which is free of rank n.
		Z is a homeomorphism of i - spaces, the i - equivariant equivalence relation ~i induces an equivalence relation (denoted by ~(i)) on the quotient Di of ix Di such that the induced map e : Di= ~(i)-!
	hopf.math.purdue.edu /Henn/sl3.txt (12658 words)

	Brian C. Hall - Department of Mathematics - University of Notre Dame
		The simplest such space is the "position Hilbert space," which is a the space of square-integrable functions over M with respect to some measure.
		In considering the quantum theory, Wren uses coherent states for the space of connections and then attempts to "project" these into the (non-existent) "gauge-invariant subspace." This "projection" is supposed to to be accomplished by integration with respect to the (also non-existent) "Haar measure" on the infinite-dimensional group of gauge transformations.
		It is thus very natural to try to reverse the roles of the compact and noncompact symmetric spaces in order to get a transform that starts with a function on a noncompact symmetric space such as hyperbolic space.
	www.nd.edu /~bhall/research (1873 words)

	MATTER and SPACE with TORSION
		Abstract: Equations are obtained describing the curvature and torsion of general metric-affine space G4 or, in accordance with the unified field theory, the distribution and motion of matter.
		Therefore, the transitional case for a spherically symmetric field will be described by complex equations, reduced from the condition related to the matter under consideration, and equations (7).
		As a result, the final solution of equations (7) in G4 space for a spherically symmetric stationary field of massless fluid with spin is given by formulae (5), (6), (11), (27).
	www.acadjournal.com /2003/v9/part4/p1 (4231 words)

	The Plancherel decomposition for a reductive symmetric space II. Representation theory - van den Ban, Schlichtkrull ...
		The Plancherel decomposition for a reductive symmetric space II.
		Abstract: We obtain the Plancherel decomposition for a reductive symmetric space in the sense of representation theory.
		van den Ban and H. Schlichtkrull, The Plancherel decomposition for a reductive symmetric space.
	citeseer.ist.psu.edu /497976.html (733 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		It is a generalisation of the notion of a symmetric space.
		Weakly symmetric spaces have a number of intriguing properties that can be explained in algebraic and geometric context.
		Classifications of weakly symmetric, commutative, and weakly commutative spaces are not known by now, but we will discuss some promising approaches to the problem.
	www.mast.queensu.ca /~cit02/talks/yakimova.html (145 words)

	What IS the Hodge Star * Map?
		If the 4-dimensional vector space of Cl(4) is Minkowski space, the two 3-dimensional bivector spaces are the Lie algebras of rotations and boosts.
		A Spin(8) bivector 2-vector space acts as a transitive transformation group of the symmetric space Spin(8) / Spin(7) = S7 and S7 x RP1 is an 8-dimensional space with octonionic structure.
		However, a Spin(8) bivector 2-vector space is too big to act as a transitive transformation group of a symmetric space of the form Spin(8) / G = M where the dimension of M is 4 or less.
	www.valdostamuseum.org /hamsmith/hstar.html (962 words)

	[No title]
		The goal of this thesis is to give more insight into the dynamics of individual isometries acting on symmetric spaces of higher rank, to describe geometrically the structure of the limit sets of discrete isometry groups, and finally to estimate their size in terms of 3 4 INTRODUCTION certain equivariant measures.
		The first two chapters are of introductory nature and provide the basics about the algebraic structure of symmetric spaces, as well as a precise description of the sphere at infinity and the Furstenberg boundary endowed with their natural topologies.
		Isomo(X) is a nonelementary discrete isometry group of a symmetric space X of noncompact type, then either the regular limit set L? \ @Xreg is empty or the set of fixed points of axial isometries is a dense subset of the limit set L?.
	www.ubka.uni-karlsruhe.de /vvv/2002/mathematik/9/9.text (9967 words)

	Is String Theory in Knots? (Site not responding. Last check: 2007-10-08)
		The symmetric group is the symmetry of fermions and bosons, while the braid group from knot theory plays the same role for anyons.
		The principle of event symmetric space-time states that the universal symmetry of physics must have a homomorphism onto the symmetric group acting on space-time events.
		There is a homomorphism from the braid group onto the symmetric group generated by the second relation.
	adela.karlin.mff.cuni.cz /~motl/Gibbs/knots.htm (2894 words)

	Poncare Dodecahedral Space and S3
		The rank of a Symmetric Space is the maximal dimension of a flat totally geodesic submanifold of the symmetric space.
		Joseph A. Wolf, The Geometry and Structure of Isotropy Irreducible Homogeneous Spaces, Acta Math.
		Therefore, S3# is a natural spinor space, and 5-fold Golden Ratio Icosahedral Symmetry is a manifestation in 3 and 4 dimensions of the Milnor sphere structure of 7 and 8 dimensions.
	www.valdostamuseum.org /hamsmith/PDS3.html (5739 words)

	AMCA: Homogeneous spaces of Iwasawa type and rank one by Maria J. Druetta (Site not responding. Last check: 2007-10-08)
		A homogeneous space of Iwasawa type and algebraic rank one is a simply connected Lie group associated to a solvable metric Lie algebra s, whose commutator n is 1-dimensional, and a unit vector H orthogonal to n can be chosen so that the restriction of ad
		In the case of Einstein spaces, it follows that the associated Lie group S is a Damek-Ricci space.
		- space of nonpositive curvature and algebraic rank one then it is a rank one symmetric space of noncompact type.
	at.yorku.ca /c/a/d/q/66.htm (378 words)

	Spectral geometry and group cohomology (Site not responding. Last check: 2007-10-08)
		This group gives rise to a locally symmetric space of (in general) infinite volume, a decomposition of the geodesic boundary of this symmetric space into a limit set and a domain of discontinuity, and to a Selberg zeta function.
		A natural question is to understand the spectral decomposition of the spaces of square integrable sections of bundles over the locally symmetric space with respect to locally invariant differential operators.
		Under additional smallness hypotheses we study the Plancherel decomposition of the Hilbert space of square integrable functions on the quotient of the semisimple group by the discrete subgroup.
	www.uni-math.gwdg.de /bunke/project1.html (364 words)

	Fulong (Site not responding. Last check: 2007-10-08)
		By analysing the temperature of two different time coordinates, we think the Hawking radiation temperature is a compensate effect under the time scale transformation.
		The line element of a spherically symmetric or plane-symmetric non-static space-time [5] is
		In this article, the difficulty of calculating energy-momentum tensors is avoided, the Hawking radiation temperature and the location of the event horizon are obtained at non-static plane-symmetric space-time and non-static spherically symmetric space-time.
	www.sif.it /cimento/tocb/112.05/02/02.html (634 words)

	Security '04 Abstract
		We postulate one such class supported by a collection of cognitive studies on visual recall, which can be characterized as mirror symmetric (reflective) passwords.
		We extend the existing analysis of graphical passwords by analyzing the size of the mirror symmetric password space relative to the full password space of the graphical password scheme of Jermyn et al.
		This reduction in size can be compensated for by longer passwords: the size of the space of mirror symmetric passwords of length about L+5 exceeds that of the full password space for corresponding length L ≤ 14 on a 5 ~ 5 grid.
	www.usenix.org /publications/library/proceedings/sec04/tech/thorpe.html (313 words)

	Polynomials, symmetry, and dynamics: An undertaking in aesthetics
		An operation that takes each point in a space A and associates it with a point in another space B is called a mapping (or map) from A to B.
		When looking for a map in a certain degree with special geometric or dynamical properties, my approach is to express the entire family of symmetric maps for that degree and, by selecting from a palette of parameters, locate a subfamily and eventually a single map with interesting behavior.
		In the first case there are four intersections of the 2-dimensional space with the superattracting 10-lines while in the second there is a single such intersection.
	www.mi.sanu.ac.yu /vismath/crass (5557 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		A quantum symmetric pair consists of the quantized enveloping algebra U for g and a left coideal subalgebra B which is a quantum analog of U(g
		The quantum symmetric space corresponding to U,B is the set of right B invariants inside the associated quantized function algebra.
		This in turn implies that the set of B bi-invariants of the quantum symmetric space is a direct sum of one-dimensional eigenspaces for the action of the center of U.
	www.ipam.ucla.edu /abstract.aspx?tid=1321 (169 words)

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