scholarly journals The Isometry Group of n-Dimensional Einstein Gyrogroup

2020 ◽  
pp. 505-512
Author(s):  
Teerapong Suksumran
Keyword(s):  
Isometry Group

scholarly journals On Polish groups admitting non-essentially countable actions

2020 ◽  
pp. 1-15
Author(s):  
ALEXANDER S. KECHRIS ◽  
MACIEJ MALICKI ◽  
ARISTOTELIS PANAGIOTOPOULOS ◽  
JOSEPH ZIELINSKI
Keyword(s):  
Equivalence Relation ◽  
Isometry Group ◽  
Alternative Proof ◽  
Orbit Equivalence ◽  
Locally Compact ◽  
Continuous Action ◽  
Polish Groups ◽  
Infinite Dimensional ◽  
Open Question ◽  
New Criterion

Abstract It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.

scholarly journals The isometry group of Outer Space

2012 ◽  
Vol 231 (3-4) ◽  
pp. 1940-1973 ◽  
Author(s):  
Stefano Francaviglia ◽  
Armando Martino
Keyword(s):  
Isometry Group ◽  
Outer Space

scholarly journals Conformal geodesics on gravitational instantons

2021 ◽  
pp. 1-32
Author(s):  
MACIEJ DUNAJSKI ◽  
PAUL TOD
Keyword(s):  
Magnetic Field ◽  
Geodesic Flow ◽  
Isometry Group ◽  
Three Dimensional ◽  
First Integrals ◽  
Complete Integrability ◽  
Conformal Killing ◽  
Gravitational Instantons ◽  
Geodesic Equations ◽  
Jacobi Equations

Abstract We study the integrability of the conformal geodesic flow (also known as the conformal circle flow) on the SO(3)–invariant gravitational instantons. On a hyper–Kähler four–manifold the conformal geodesic equations reduce to geodesic equations of a charged particle moving in a constant self–dual magnetic field. In the case of the anti–self–dual Taub NUT instanton we integrate these equations completely by separating the Hamilton–Jacobi equations, and finding a commuting set of first integrals. This gives the first example of an integrable conformal geodesic flow on a four–manifold which is not a symmetric space. In the case of the Eguchi–Hanson we find all conformal geodesics which lie on the three–dimensional orbits of the isometry group. In the non–hyper–Kähler case of the Fubini–Study metric on $\mathbb{CP}^2$ we use the first integrals arising from the conformal Killing–Yano tensors to recover the known complete integrability of conformal geodesics.

scholarly journals NEW SUPERSTRING ISOMETRIES AND HIDDEN DIMENSIONS

2007 ◽  
Vol 22 (29) ◽  
pp. 5301-5323 ◽  
Author(s):  
DIMITRI POLYAKOV
Keyword(s):  
Extra Dimension ◽  
Isometry Group ◽  
Space Time ◽  
Time Dimension ◽  
Superstring Theory ◽  
Symmetry Transformations ◽  
Noncritical Strings ◽  
A Chain ◽  
Symmetry Generators ◽  
Special Space

We study the hierarchy of hidden space–time symmetries of noncritical strings in RNS formalism, realized nonlinearly. Under these symmetry transformations the variation of the matter part of the RNS action is canceled by that of the ghost part. These symmetries, referred to as the α-symmetries, are induced by special space–time generators, violating the equivalence of ghost pictures. We classify the α-symmetry generators in terms of superconformal ghost cohomologies Hn ~ H-n-2(n≥0) and associate these generators with a chain of hidden space–time dimensions, with each ghost cohomology Hn ~ H-n-2 "contributing" an extra dimension. Namely, we show that each ghost cohomology Hn ~ H-n-2 of noncritical superstring theory in d-dimensions contains d+n+1 α-symmetry generators and the generators from Hk ~ H-k-2, 1≤k ≤n, combined together, extend the space–time isometry group from the naive SO (d, 2) to SO (d+n, 2). In the simplest case of n = 1 the α-generators are identified with the extra symmetries of the 2T-physics formalism, also known to originate from a hidden space–time dimension.

scholarly journals Positively curved 6-manifolds with simple symmetry groups

2002 ◽  
Vol 74 (4) ◽  
pp. 589-597 ◽  
Author(s):  
FUQUAN FANG
Keyword(s):  
Sectional Curvature ◽  
Isometry Group ◽  
Symmetry Groups ◽  
Positive Sectional Curvature ◽  
Identity Component ◽  
Simply Connected ◽  
Lie Subgroup ◽  
Positively Curved

Let M be a simply connected compact 6-manifold of positive sectional curvature. If the identity component of the isometry group contains a simple Lie subgroup, we prove that M is diffeomorphic to one of the five manifolds listed in Theorem A.

scholarly journals Projective properties of Lorentzian surfaces

2019 ◽  
pp. 1-13
Author(s):  
Pierre Mounoud
Keyword(s):  
Infinite Order ◽  
Isometry Group ◽  
Projective Transformation ◽  
Compact Surface ◽  
Projective Group ◽  
Flat Torus

We investigate projective properties of Lorentzian surfaces. In particular, we prove that if T is a non-flat torus, then the index of its isometry group in its projective group is at most two. We also prove that any topologically finite non-compact surface can be endowed with a metric having a non-isometric projective transformation of infinite order.

scholarly journals Non-Abelian U -duality for membranes

2020 ◽  
Vol 2020 (7) ◽  
Author(s):  
Yuho Sakatani ◽  
Shozo Uehara
Keyword(s):  
String Theory ◽  
Isometry Group ◽  
Standard Treatment ◽  
Abelian Extension ◽  
Target Space ◽  
Membrane Theory ◽  
Drinfel’D Double ◽  
M Theory

Abstract The $T$-duality of string theory can be extended to the Poisson–Lie $T$-duality when the target space has a generalized isometry group given by a Drinfel’d double. In M-theory, $T$-duality is understood as a subgroup of $U$-duality, but the non-Abelian extension of $U$-duality is still a mystery. In this paper we study membrane theory on a curved background with a generalized isometry group given by the $\mathcal E_n$ algebra. This provides a natural setup to study non-Abelian $U$-duality because the $\mathcal E_n$ algebra has been proposed as a $U$-duality extension of the Drinfel’d double. We show that the standard treatment of Abelian $U$-duality can be extended to the non-Abelian setup. However, a famous issue in Abelian $U$-duality still exists in the non-Abelian extension.

scholarly journals Conjugate Reciprocal Polynomials with All Roots on the Unit Circle

2008 ◽  
Vol 60 (5) ◽  
pp. 1149-1167 ◽  
Author(s):  
Kathleen L. Petersen ◽  
Christopher D. Sinclair
Keyword(s):  
Unit Circle ◽  
Lebesgue Measure ◽  
Dihedral Group ◽  
Isometry Group ◽  
Fixed Degree

AbstractWe study the geometry, topology and Lebesgue measure of the set of monic conjugate reciprocal polynomials of fixed degree with all roots on the unit circle. The set of such polynomials of degree N is naturally associated to a subset of ℝN−1. We calculate the volume of this set, prove the set is homeomorphic to the N − 1 ball and that its isometry group is isomorphic to the dihedral group of order 2N.

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