scholarly journals Cauchy Symmetric Spaces with Point-countable cs-Networks

2016 ◽  
Vol 57 (1) ◽  
pp. 163-170
Author(s):  
Luong Quoc Tuyen
Keyword(s):  
Symmetric Space ◽  
Metric Space ◽  
Symmetric Spaces

Abstract In this paper, we prove that a Cauchy symmetric space has a point-countable cs-network if and only if it is a 1-sequence-covering compact-covering quotient π, s-image of a metric space; if and only if it is a sequence-covering quotient π, s-image of a metric space.

scholarly journals THE SMOOTHNESS OF ORBITAL MEASURES ON NONCOMPACT SYMMETRIC SPACES

2021 ◽  
pp. 1-20
Author(s):  
SANJIV KUMAR GUPTA ◽  
KATHRYN E. HARE
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Continuous Measure ◽  
Convolution Product ◽  
Absolutely Continuous ◽  
Regular Elements ◽  
Connected Subgroup ◽  
Decay Properties ◽  
Absolutely Continuous Measure ◽  
Special Case

Abstract Let $G/K$ be an irreducible symmetric space, where G is a noncompact, connected Lie group and K is a compact, connected subgroup. We use decay properties of the spherical functions to show that the convolution product of any $r=r(G/K)$ continuous orbital measures has its density function in $L^{2}(G)$ and hence is an absolutely continuous measure with respect to the Haar measure. The number r is approximately the rank of $G/K$ . For the special case of the orbital measures, $\nu _{a_{i}}$ , supported on the double cosets $Ka_{i}K$ , where $a_{i}$ belongs to the dense set of regular elements, we prove the sharp result that $\nu _{a_{1}}\ast \nu _{a_{2}}\in L^{2},$ except for the symmetric space of Cartan class $AI$ when the convolution of three orbital measures is needed (even though $\nu _{a_{1}}\ast \nu _{a_{2}}$ is absolutely continuous).

Corwin–Greenleaf multiplicity functions for complex semisimple symmetric spaces

2015 ◽  
Vol 26 (05) ◽  
pp. 1550039
Author(s):  
Salma Nasrin
Keyword(s):  
Symmetric Space ◽  
Lie Algebras ◽  
Lie Group ◽  
Symmetric Spaces ◽  
Coadjoint Orbit ◽  
Natural Projection ◽  
Real Form ◽  
Single Orbit ◽  
Complex Semisimple ◽  
Complex Symmetric

Let Gℂ be a complex simple Lie group, GU a compact real form, and [Formula: see text] the natural projection between the dual of the Lie algebras. We prove that, for any coadjoint orbit [Formula: see text] of GU, the intersection of [Formula: see text] with a coadjoint orbit [Formula: see text] of Gℂ is either an empty set or a single orbit of GU if [Formula: see text] is isomorphic to a complex symmetric space.

DECAY RATES FOR THE SHIFTED WAVE EQUATION ON A SYMMETRIC SPACE OF NONCOMPACT TYPE

2013 ◽  
Vol 10 (04) ◽  
pp. 677-701
Author(s):  
CARLOS ALMADA
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Symmetric Space ◽  
Decay Rate ◽  
Wave Operator ◽  
Symmetric Spaces ◽  
Decay Rates ◽  
Noncompact Type ◽  
Killing Fields ◽  
The Cauchy Problem

We derive L∞–L1 decay rate estimates for solutions of the shifted wave equation on certain symmetric spaces (M, g). The Cauchy problem for the shifted wave operator on these spaces was studied by Helgason, who obtained a closed form for its solution. Our results extend to this new context the classical estimates for the wave equation in ℝn. Then, following an idea from Klainerman, we introduce a new norm based on Lie derivatives with respect to Killing fields on M and we derive an estimate for the case that n = dim M is odd.

scholarly journals On projective-symmetric spaces

1964 ◽  
Vol 4 (1) ◽  
pp. 113-121 ◽  
Author(s):  
Bandana Gupta
Keyword(s):  
Symmetric Space ◽  
Curvature Tensor ◽  
Symmetric Spaces ◽  
Covariant Differentiation ◽  
Metric Tensor ◽  
Projective Curvature ◽  
Projective Curvature Tensor ◽  
Image Position ◽  
Riemannian Spaces

This paper deals with a type of Remannian space Vn (n ≧ 2) for which the first covariant dervative of Weyl's projective curvature tensor is everywhere zero, that is where comma denotes covariant differentiation with respect to the metric tensor gij of Vn. Such a space has been called a projective-symmetric space by Gy. Soós [1]. We shall denote such an n-space by ψn. It will be proved in this paper that decomposable Projective-Symmetric spaces are symmetric in the sense of Cartan. In sections 3, 4 and 5 non-decomposable spaces of this kind will be considered in relation to other well-known classes of Riemannian spaces defined by curvature restrictions. In the last section the question of the existence of fields of concurrent directions in a ψ will be discussed.

CLASSES MINIMALES DE RÉSEAUX ET RÉTRACTIONS GÉOMÉTRIQUES ÉQUIVARIANTES DANS LES ESPACES SYMÉTRIQUES

2001 ◽  
Vol 64 (2) ◽  
pp. 275-286 ◽  
Author(s):  
CHRISTOPHE BAVARD
Keyword(s):  
Finite Group ◽  
Euclidean Space ◽  
Symmetric Space ◽  
Group Action ◽  
Symmetric Spaces ◽  
Classification Problem ◽  
Finite Group Action ◽  
Natural Geometry ◽  
A Finite Group

Equivariant and cocompact retractions of certain symmetric spaces are constructed. These retractions are defined using the natural geometry of symmetric spaces and in relation to the theory of lattices of euclidean space. The following cases are considered: the symmetric space corresponding to lattices endowed with a finite group action, from which is obtained some information relating to the classification problem of these lattices, and the Siegel space Sp2g(R)/Ug, for which a natural Sp2g(Z)-equivariant cocompact retract of codimension 1 is obtained.

scholarly journals Vassiliev invariants from symmetric spaces

2016 ◽  
Vol 25 (10) ◽  
pp. 1650055 ◽  
Author(s):  
Indranil Biswas ◽  
Niels Leth Gammelgaard
Keyword(s):  
Symmetric Space ◽  
Lie Algebra ◽  
Curvature Tensor ◽  
Symmetric Spaces ◽  
Weight System ◽  
Riemannian Symmetric Space ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Vassiliev Invariants ◽  
Chord Diagrams

We construct a natural framed weight system on chord diagrams from the curvature tensor of any pseudo-Riemannian symmetric space. These weight systems are of Lie algebra type and realized by the action of the holonomy Lie algebra on a tangent space. Among the Lie algebra weight systems, they are exactly characterized by having the symmetries of the Riemann curvature tensor.

INTEGRABILITY AND SYMMETRIC SPACES II: THE COSET SPACES

1989 ◽  
Vol 04 (03) ◽  
pp. 675-699 ◽  
Author(s):  
L. A. FERREIRA
Keyword(s):  
Symmetric Space ◽  
Poisson Bracket ◽  
Algebraic Structure ◽  
Symmetric Spaces ◽  
Sufficient Condition ◽  
Conserved Quantities ◽  
Coset Space ◽  
Canonical Variables ◽  
Coset Spaces

It is shown that a sufficient condition for a model describing the motion of a particle on a coset space to possess a Fundamental Poisson bracket Relation, and consequently charges in involution, is that it must be a symmetric space. The conditions, a Hamiltonian, or any functions of the canonical variables, has to satisfy in order to commute with these charges, are studied. It is shown that, for the case of the noncompact symmetric spaces, these conditions lead to an algebraic structure which plays an important role in the construction of conserved quantities.

Some Hypersurfaces of Symmetric Spaces

1983 ◽  
Vol 26 (3) ◽  
pp. 303-311 ◽  
Author(s):  
Yoshio Matsuyama
Keyword(s):  
Symmetric Space ◽  
Projective Space ◽  
Symmetric Spaces ◽  
Real Projective Space ◽  
Principal Curvatures

AbstractIn this paper we consider how much we can say about an irreducible symmetric space M which admits a hypersurface N with at most two distinct principal curvatures. Then we will obtain that (1) if N is locally symmetric, then M must be a sphere, a real projective space and their noncompact duals (2) if N is Einstein, then M must be rank 1.

On an Upper Bound for the Heat Kernel on SU*(2n)/ Sp(n)

1994 ◽  
Vol 37 (3) ◽  
pp. 408-418 ◽  
Author(s):  
P. Sawyer
Keyword(s):  
Symmetric Space ◽  
Heat Kernel ◽  
Upper Bound ◽  
Symmetric Spaces ◽  
Legendre Function ◽  
Noncompact Type ◽  
Image Position

AbstractJean-Philippe Anker made an interesting conjecture in [2] about the growth of the heat kernel on symmetric spaces of noncompact type. For any symmetric space of noncompact type, we can writewhere ϕ0 is the Legendre function and q, "the dimension at infinity", is chosen such that limt—>∞Vt(x) = 1 for all x. Anker's conjecture can be stated as follows: there exists a constant C > 0 such thatwhere is the set of positive indivisible roots. The behaviour of the function ϕ0 is well known (see [1]).The main goal of this paper is to establish the conjecture for the spaces SU*(2n)/ Sp(n).

scholarly journals On the index of symmetric spaces

2018 ◽  
Vol 2018 (737) ◽  
pp. 33-48 ◽  
Author(s):  
Jürgen Berndt ◽  
Carlos Olmos
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Riemannian Symmetric Space ◽  
Totally Geodesic Submanifold ◽  
Totally Geodesic ◽  
Geodesic Submanifold ◽  
Riemannian Symmetric Spaces

AbstractLetMbe an irreducible Riemannian symmetric space. The index ofMis the minimal codimension of a (nontrivial) totally geodesic submanifold ofM. We prove that the index is bounded from below by the rank of the symmetric space. We also classify the irreducible Riemannian symmetric spaces whose index is less than or equal to 3.

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