scholarly journals Continuous Hahn Polynomials of Differential Operator Argument and Analysis on Riemannian Symmetric Spaces of Constant Curvature

1992 ◽  
Vol 44 (4) ◽  
pp. 750-773 ◽  
Author(s):  
Erich Badertscher ◽  
Tom H. Koornwinder
Keyword(s):  
Differential Operator ◽  
Orthogonal Polynomials ◽  
Constant Curvature ◽  
Symmetric Spaces ◽  
Spherical Functions ◽  
Spaces Of Constant Curvature ◽  
Hahn Polynomials ◽  
Riemannian Symmetric Spaces ◽  
Simply Connected ◽  
Continuous Hahn Polynomials

AbstractFor the three types of simply connected Riemannian spaces of constant curvature it is shown that the associated spherical functions can be obtained from the corresponding (zonal) spherical functions by application of a differential operator of the form p(i d/dt), where p belongs to a system of orthogonal polynomials: Gegenbauer polynomials, Hahn polynomials or continuous symmetric Hahn polynomials. We give a group theoretic explanation of this phenomenon and relate the properties of the polynomials p to the properties of the corresponding representation. The method is extended to the case of intertwining functions.

scholarly journals The classification of simply-connected contact sub-riemannian symmetric spaces

1999 ◽  
Vol 188 (1) ◽  
pp. 65-82 ◽  
Author(s):  
Pierre Bieliavsky ◽  
Elisha Falbel ◽  
Claudio Gorodski
Keyword(s):  
Symmetric Spaces ◽  
Riemannian Symmetric Spaces ◽  
Simply Connected

scholarly journals Special semi-reducible pseudo-Riemannian spaces

2021 ◽  
Vol 14 (1) ◽  
pp. 48-59
Author(s):  
Юлія Степанівна Федченко ◽  
Олександр Васильович Лесечко
Keyword(s):  
Constant Curvature ◽  
Symmetric Spaces ◽  
Necessary Conditions ◽  
Conformally Flat ◽  
Spaces Of Constant Curvature ◽  
Riemannian Spaces

The paper contains necessary conditions allowing to reduce matrix tensors of pseudo-Riemannian spaces to special forms called semi-reducible, under assumption that the tensor defining tensor characteristic of semireducibility spaces, is idempotent. The tensor characteristic is reduced to the spaces of constant curvature, Ricci-symmetric spaces and conformally flat pseudo-Riemannian spaces. The obtained results can be applied for construction of examples of spaces belonging to special types of pseudo-Riemannian spaces. The research is carried out locally in tensor shape, without limitations imposed on a sign of a metric.  

scholarly journals Harmonic morphisms from solvable Lie groups

2009 ◽  
Vol 147 (2) ◽  
pp. 389-408 ◽  
Author(s):  
SIGMUNDUR GUDMUNDSSON ◽  
MARTIN SVENSSON
Keyword(s):  
Lie Groups ◽  
Symmetric Spaces ◽  
Global Solutions ◽  
Riemannian Symmetric Space ◽  
Continuous Family ◽  
Harmonic Morphisms ◽  
Solvable Lie Groups ◽  
3 Dimensional ◽  
Riemannian Symmetric Spaces ◽  
Simply Connected

AbstractIn this paper we introduce two new methods for constructing harmonic morphisms from solvable Lie groups. The first method yields global solutions from any simply connected nilpotent Lie group and from any Riemannian symmetric space of non-compact type and rank r ≥ 3. The second method provides us with global solutions from any Damek–Ricci space and many non-compact Riemannian symmetric spaces. We then give a continuous family of 3-dimensional solvable Lie groups not admitting any complex-valued harmonic morphisms, not even locally.

An extension theorem for non-compact split embedded Riemannian symmetric spaces and an application to their universal property

2020 ◽  
Vol 20 (4) ◽  
pp. 499-506
Author(s):  
Julius Grüning ◽  
Ralf Köhl
Keyword(s):  
Symmetric Space ◽  
Central Extension ◽  
Hyperbolic Plane ◽  
Symmetric Spaces ◽  
Extension Theorem ◽  
Universal Property ◽  
Riemannian Symmetric Space ◽  
Differential Structure ◽  
Riemannian Symmetric Spaces ◽  
Simply Connected

AbstractBy [5] it is known that a geodesic γ in an abstract reflection space X (in the sense of Loos, without any assumption of differential structure) canonically admits an action of a 1-parameter subgroup of the group of transvections of X. In this article, we modify these arguments in order to prove an analog of this result stating that, if X contains an embedded hyperbolic plane 𝓗 ⊂ X, then this yields a canonical action of a subgroup of the transvection group of X isomorphic to a perfect central extension of PSL2(ℝ). This result can be further extended to arbitrary Riemannian symmetric spaces of non-compact split type Y lying in X and can be used to prove that a Riemannian symmetric space and, more generally, the Kac–Moody symmetric space G/K for an algebraically simply connected two-spherical split Kac–Moody group G, as defined in [5], satisfies a universal property similar to the universal property that the group G satisfies itself.

Center of Mass Theorem and the Separation of Variables for Two-Body System on a Three-Dimensional Sphere

2020 ◽  
Vol 23 (3) ◽  
pp. 306-311
Author(s):  
Yu. Kurochkin ◽  
Dz. Shoukavy ◽  
I. Boyarina
Keyword(s):  
Constant Curvature ◽  
Jacobi Equation ◽  
Separation Of Variables ◽  
Center Of Mass ◽  
Three Dimensional ◽  
Relative Distance ◽  
Dimensional Sphere ◽  
Body System ◽  
Spaces Of Constant Curvature ◽  
Hamilton Jacobi Equation

The immobility of the center of mass in spaces of constant curvature is postulated based on its definition obtained in [1]. The system of two particles which interact through a potential depending only on the distance between particles on a three-dimensional sphere is considered. The Hamilton-Jacobi equation is formulated and its solutions and trajectory equations are found. It was established that the reduced mass of the system depends on the relative distance.

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