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Topic: Isometry group

	ISOMETRY GROUP : Encyclopedia Entry
		In mathematics, the isometry group of a metric space is the set of all isometries from the metric space onto itself, with the function composition as group operation.
		An isometry group of a metric space is a subgroup of isometries; it represents in most cases a possible set of symmetries of objects/figures in the space, or functions defined on the space.
		Then, the isometry group of the set of three vertices of this triangle is the trivial group.
	www.bibleocean.com /OmniDefinition/Isometry_group (159 words)

	Symmetry group - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-30)
		It is a subgroup of the isometry group of the space concerned.
		The group of all symmetries of a sphere O(3) is an example of this, and in general such continuous symmetry groups are studied as Lie groups.
		With a categorization of subgroups of the Euclidean group corresponds a categorization of symmetry groups.
	en.wikipedia.org /wiki/Symmetry_group (1622 words)

	Isometry -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-30)
		Isometries are often used in constructions where one space is (additional info and facts about embedded) embedded in another space.
		A path isometry or arcwise isometry is a map which preserves the (additional info and facts about lengths of curves) lengths of curves (not necessarily bijective).
		In algebraic terms the isometries form a group called (additional info and facts about Euclidean group) Euclidean group which is the (additional info and facts about semidirect product) semidirect product of the (additional info and facts about orthogonal group) orthogonal group and the group of translations.
	www.absoluteastronomy.com /encyclopedia/i/is/isometry.htm (466 words)

	Automorphism - Wikipedia, the free encyclopedia
		A group automorphism is a group isomorphism from a group to itself.
		In the case of a Galois extension L/K the subgroup of all automorphisms of L fixing K pointwise is called the Galois group of the extension.
		The inner automorphisms are the conjugations by the elements of the group itself.
	en.wikipedia.org /wiki/Automorphism (875 words)

	PlanetMath: isometry
		An isometric mapping that is surjective (and therefore bijective) is called an isometry.
		In general, an (as opposed to the) isometry group (or group of isometries) of
		This is version 10 of isometry, born on 2002-02-13, modified 2006-12-15.
	www.planetmath.org /encyclopedia/Isometry.html (179 words)

	PlanetMath: classical groups
		It is commonplace to express the classical groups with explicit matrices; however, the theory and classification of classical groups can benefit from a basis free consideration.
		Because symplectic spaces have a standard hyperbolic basis it follows every symplectic group over a vector space of the same dimension is isometric, meaning isomorphic as vector spaces but with an isomorphism which respects the forms.
		This is version 20 of classical groups, born on 2006-04-08, modified 2007-06-23.
	planetmath.org /encyclopedia/Isometry2.html (556 words)

	Automorphism Group and Isometry Testing
		This function computes the automorphism group G of a lattice L which is defined to be the group of those automorphisms of the Z-module underlying L which preserve the inner product of L. L must be an exact lattice (over Z or Q).
		G does not act on the elements of L, since there is no natural matrix action of the automorphism group on L in the case that the rank of L is less than its degree.
		This function computes the subgroup of the automorphism group of L which fixes also the forms given in the set or sequence F. The matrices in F are not required to be positive definite or even symmetric.
	www.umich.edu /~gpcc/scs/magma/text813.htm (1543 words)

	Isometry (Site not responding. Last check: 2007-10-30)
		In the mathematical discipline of geometry and mathematical analysis, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces.
		Isometries are often used in constructions where one space is embedded in another space.
		In algebraic terms the isometries form a group called Euclidean group which is the semidirect product of the orthogonal group and the group of translations.
	www.worldhistory.com /wiki/I/Isometry.htm (421 words)

	[No title] (Site not responding. Last check: 2007-10-30)
		Actions, a garden of G-spaces [3-dimensional rotation group centered on origin (orbits are spheres) matrix groups acting on vector spaces, Euclidean group Euclidean (n)={x maps to Ax+b: A orthogonal} Affine group AGL(n,R)={x maps to Ax+b: A invertible} Rotation group of a cube.
		The Euclidean and similarity groups acting on triangles of R^2 : congruence and similarity of triangles; hence congruence and similarity of configurations of the real plane defined to be being in the same orbit under Euclidean/similarity group, respectively.
		Classification of finite groups of isometries in 2 and 3 dimensions over the reals as subgroups of the stabilisers of regular polygons and regular polyhedra.
	www.maths.uwa.edu.au /students/outlines/97S1/3P5.php (311 words)

	Euclidean group (Site not responding. Last check: 2007-10-30)
		In mathematics, the Euclidean group is the symmetry group associated with Euclidean geometry.
		Writing E(n) for the Euclidean group of symmetries of n-dimensional Euclidean space, it may also be described as the isometry group of the Euclidean metric.
		That is, isometries not involving a change of orientation; equally, those represented as a translation followed by a rotation, rather than a translation followed by some kind of reflection (in dimensions 2 and 3, these are the familiar reflections in a mirror line or plane, which may be taken to include the origin).
	www.worldhistory.com /wiki/E/Euclidean-group.htm (500 words)

	[No title] (Site not responding. Last check: 2007-10-30)
		The isometry group of a compact Lorentz manifold, I, with Garrett Stuck, {\it Inventiones Mathematic\ae}, {\bf129} (1997), 239-261.
		The isometry group of a compact Lorentz manifold, II, with Garrett Stuck, {\it Inventiones Mathematic\ae}, {\bf129} (1997), 263-287.
		For example, one group may be $\splin_2(\R)$ and the other a group with infinite discrete center ({\it e.g.}, the universal cover of $\splin_2(\R)$); I believe this is the first rigidity result of this type for a pair of simple Lie groups both of split rank one.
	www.math.umn.edu /~adams/publ.txt (2431 words)

	Construction of configuration space geometry (Site not responding. Last check: 2007-10-30)
		The group theoretical construction of the configuration space geometry is based on the miraculous properties possessed by the boundary of the 4-dimensional lightcone.
		The correct identification for the isometry group of C(H) turns out to be as the group Can(delta H) of the canonical transformations of delta H= delta M^4_+ xCP_2: any function of delta H coordinates serves as a Hamiltonian of some canonical transformation.
		Or equivalently, the group of delta H canonical transformations is the group of canonical transformations of S^2xCP_2 localized with respect to the radial coordinate of the light cone boundary.
	www.physics.helsinki.fi /~matpitka/geomc.html (591 words)

	Automorphism - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-10-30)
		In group theory, for example, let a be an element of a group G.
		Conjugation by a is the group homomorphism φ
		The quotient group Aut(G) / Inn(G) is usually denoted by Out(G).
	encyclopedia.learnthis.info /a/au/automorphism_1.html (849 words)

	Automorphism Group and Isometry Testing
		This function computes the automorphism group G of a lattice L which is defined to be the group of those automorphisms of the Z-module underlying L which preserve the inner product of L. L must be an exact lattice (over Z or Q).
		This function computes the subgroup of the automorphism group of L which fixes also the forms given in the set or sequence F. The matrices in F are not required to be positive definite or even symmetric.
		In this example we compute the automorphism group of the root lattice E_8 and manually transform the action on the coordinates into an action on the lattice vectors.
	www.math.wisc.edu /help/magma/text549.html (1543 words)

	Enumeration of 2-isohedral tilings on the sphere (Site not responding. Last check: 2007-10-30)
		These groups may be considered as 3-dimensional isometry groups with an invariant point - the centre of the sphere - and are well known.
		The enumeration of these groups is given in [7], but in this paper we prefer to use for them other symbols - orbifold symbols [10] - which reflect their nature as isometry groups of a sphere.
		Numerical data of the obtained Delone classes of isohedral tilings of the sphere with disks for all series and isometry groups coincide with numerical data of work [8].
	www.mi.sanu.ac.yu /vismath/zamorzaeva (1851 words)

	Professor Stephen Hawking
		In these, the U1 isometry group can have fixed points on surfaces of any even co-dimension, and the space-time need not be asymptotically flat, or asymptotically anti de Sitter.
		The situation is very different however, if the solution can't be foliated by surfaces of constant tau, where tau is the parameter of the U1 isometry group, which agrees with the periodic identification at infinity.
		One of these is that, because the spatial rotation group, O4, is of rank 2, there are two rotation parameters, a and b.
	www.hawking.org.uk /text/physics/nut.html (4930 words)

	[No title] (Site not responding. Last check: 2007-10-30)
		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.
		Since the isometry group does not act transitively on the geometric boundary, it is not possible that a sequence of axial isometries maps all of the geometric boundary to the limit of its attractive fixed points.
		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)

	Configuration space geometry (Site not responding. Last check: 2007-10-30)
		The existence of a Riemann connection requires in the infinite-dimensional case that the group of the configuration space isometries is infinite-dimensional and in a well defined sense maximal.
		Label "i" corresponds to zero modes, that is variables which are isometry invariants or do not appear as coordinate differentials in the configuration space metric.
		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)

	[No title]
		(This is true for all simple Lie groups, by a result of Whitehead.) The 3rd homotopy group of E8 is Z. (This is true for all simple Lie groups, by another result of Whitehead.) The 4th homotopy group of E8 is trivial.
		Still, there is a nice Riemannian manifold whose isometry group is E8 and whose dimension is 2 x 8 x 8 = 128, exactly right for being the octooctonionic projective plane...
		So at the group level we can form the quotient E8/Spin(16) and get a homogeneous space whose isometry group is E8 and whose dimension is that of S+, namely 128.
	www.math.niu.edu /~rusin/known-math/00_incoming/E8 (763 words)

	Exceptional Lie Algebras
		as isometries of the sphere yields an action of this group as isometries of
		The Lie algebra of this isometry group is
		It thus inherits a Riemannian metric from this sphere, and the unitary group
	math.ucr.edu /home/baez/octonions/node13.html (512 words)

	[No title]
		Groups can also act as symmetries of other groups, and in combination with Sylow's theorems, this gives a powerful way of constructing groups.
		To display and exemplify the ubiquity of groups as symmetries of physical and mathematical objects.
		Group of isometries and groups of translations of R
	www.shef.ac.uk /nps/courses/groups/info.html (272 words)

	My Personal Reading List
		The Ernst equation for gravitational fields with a two-parameter isometry group is formulated as a vanishing-curvature condition on an SU(2) or SU(1,1) bundle, both in the elliptic and hyperbolic cases.
		The equations for a gravitational field in the presence of a two-parameter Abelian group of isometries (e.g., stationary axisymmetric fields or cylindrical waves) are shown to be equivalent to the construction of a connection (with vanishing curvature) on a principal bundle with an appropriate structure group.
		We identify the Geroch group (Kinnersley and Chitre's group K) with a subgroup of the loop group of GL(2,C) and we describe its orbits.
	members.localnet.com /~atheneum/bib/staxsym.html (14202 words)

	A Visualization of the Isometry Group Action on the Fomenko-Matveev-Weeks Manifold (ResearchIndex) (Site not responding. Last check: 2007-10-30)
		It is known, that the isometry group of the manifold g/t is isomorphic to the dihedral group ID6 of order 12.
		We study the lattice of the action of the isometry group on the manifold A/t1 and obtain all orbifolds which arised as the quotient spaces.
		4 The fundamental group of the orbit space of a discontinuous..
	citeseer.ist.psu.edu /514355.html (240 words)

	[No title] (Site not responding. Last check: 2007-10-30)
		========================================================================== ABSTRACTS Mikhail Belolipetsky Finite groups and hyperbolic manifolds The isometry group of a compact n-dimensional hyperbolic manifold is known to be finite.
		By classifying exceptional pairs we establish that a group is almost always uniquely determined by its invariants, for other classes of groups, namely, irreducible and orthogonal groups.
		Eugene Plotkin Engel-like characterization of radicals in finite groups and finite dimensional Lie algebras The following theorem by R.Baer is well-known: in any finite group its nilpotent radical coincides with the set of all Engel elements.
	www.math.technion.ac.il /~techm/20040623000020040624ami (635 words)

	Volume 08 Abstracts (Site not responding. Last check: 2007-10-30)
		It is known, that the isometry group of the manifold M
		Furthermore, it is shown that in case L is connected, it is reductive and its semi-simple Levi compontent E acts transitively on B(S), which turns out to be a flag manifold of E, as well.
		This article relates the Bruhat-Chevalley order in the Weyl group W of G to the ordering of the control sets for S in the flag manifolds of G by showing that the one-to-one correspondence between the control sets and the elements of a couble coset of W reverses the orders.
	www.heldermann.de /JLT/jltabs08.htm (971 words)

	[No title]
		Generators of cyclic groups Properties: any cyclic group is isomorphic to Z or Zn.
		G/H is a group (proof); - Th.9.3 G/Z(G) cyclic => G Abelian (proof); - Th.9.4 G/Z(G) (Inn(G) (representation of G by inner conjugation) (proof) (use 1st Iso.
		27 Symmetry groups - Isometry, symmetry group of a figure - Translations, rotations, reflections, glyde-reflections - Structure of isometries of Rn - Classification of finite plane symmetries and groups of rotations in R3.
	www.ilstu.edu /~lmiones/336rvs05.DOC (846 words)

	Riemannian geometry (Site not responding. Last check: 2007-10-30)
		There is a constant C=C(n) such that if M is a compact connected n-dimensional Riemannian manifold with positive sectional curvature then sum of its Betti numbers is at most C.
		If M is a complete Riemannian manifold with negative sectional curvature then any abelian subgroup of its fundamental group of M is isomorphic to Z.
		Any compact Riemannian manifold with negative Ricci curvature has discrete isometry group.
	www.brainyencyclopedia.com /encyclopedia/r/ri/riemannian_geometry.html (775 words)

	Nilpotent Lie Groups (Site not responding. Last check: 2007-10-30)
		In particular, we integrate the geodesic equation, discuss the structure of the isometry group, and make a study of lattices and periodic geodesics.
		While still a semidirect product, the isometry group can be strictly larger than the obvious analogue.
		We present examples of geodesics, geodesic surfaces, and conjugate loci in the 3-dimensional Heisenberg group with signatures (+ - +) and (- - +).
	www.math.wichita.edu /~pparker/research/prnlg.htm (434 words)

	Su Gao: papers (Site not responding. Last check: 2007-10-30)
		We characterize the complexity of the classification problem of Polish metric spaces up to isometry as, up to Borel bireducibility, the unique univeral orbit equivalence relations induced by Borel actions of Polish groups.
		As a corollary of the proof, we derive that every Polish group is isomorphic to the isometry group of some Polish metric space.
		We also develop an analysis of the isometry groups of locally compact separable metric spaces and give a complete characterization for this class of Polish groups.
	www.cco.caltech.edu /~sugao/paper10.html (142 words)

	[No title] (Site not responding. Last check: 2007-10-30)
		(Incidently, whenever we have an isometry group acting on a spacetime, we can define the isotropy subgroup to be the subgroup which does not move a particular event in any orbit.
		Then, the dimension of the isometry group minus the dimension of the isotropy subgroup equals the dimension of the orbit.
		In the case of the Poincare group, the isotropy subgroup is SO(1,3), and 10-6 = 4, so the orbit is four dimensional, in fact the entire spacetime.
	math.ucr.edu /home/baez/PUB/einsteinmaxwell (2953 words)

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