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.

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.

scholarly journals EXTERIOR DIFFERENTIAL FORMS ON RIEMANNIAN SYMMETRIC SPACES

2017 ◽  
pp. 49-53
Author(s):  
Irina Alexandrova ◽  
Irina Alexandrova ◽  
Sergey Stepanov ◽  
Sergey Stepanov ◽  
Irina Tsyganok ◽  
...  
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Differential Forms ◽  
Conformal Killing ◽  
Riemannian Symmetric Space ◽  
Vanishing Theorems ◽  
Harmonic Forms ◽  
Exterior Differential ◽  
Rank One ◽  
Riemannian Symmetric Spaces

In the present paper we give a rough classification of exterior differential forms on a Riemannian manifold. We define conformal Killing, closed conformal Killing, coclosed conformal Killing and harmonic forms due to this classification and consider these forms on a Riemannian globally symmetric space and, in particular, on a rank-one Riemannian symmetric space. We prove vanishing theorems for conformal Killing L 2-forms on a Riemannian globally symmetric space of noncompact type. Namely, we prove that every closed or co-closed conformal Killing L 2-form is a parallel form on an arbitrary such manifold. If the volume of it is infinite, then every closed or co-closed conformal Killing L 2-form is identically zero. In addition, we prove vanishing theorems for harmonic forms on some Riemannian globally symmetric spaces of compact type. Namely, we prove that all harmonic one-formsvanish everywhere and every harmonic r -form  r  2 is parallel on an arbitrary such manifold. Our proofs are based on the Bochnertechnique and its generalized version that are most elegant and important analytical methods in differential geometry “in the large”.

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.

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.

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 Beurling's Theorem and Characterization of Heat Kernel for Riemannian Symmetric Spaces of Noncompact Type

2007 ◽  
Vol 50 (2) ◽  
pp. 291-312 ◽  
Author(s):  
Rudra P. Sarkar ◽  
Jyoti Sengupta
Keyword(s):  
Symmetric Space ◽  
Heat Kernel ◽  
Symmetric Spaces ◽  
Noncompact Type ◽  
Beurling’S Theorem ◽  
Riemannian Symmetric Spaces

AbstractWe prove Beurling's theorem for rank 1 Riemannian symmetric spaces and relate its consequences with the characterization of the heat kernel of the symmetric space.

scholarly journals A compact imbedding of Riemannian Symmetric Spaces

2018 ◽  
Vol 127 (1A) ◽  
pp. 55
Author(s):  
Trần Đạo Dõng
Keyword(s):  
Symmetric Space ◽  
Root System ◽  
Symmetric Spaces ◽  
Compact Subgroup ◽  
Analytic Structure ◽  
Riemannian Symmetric Space ◽  
The Real ◽  
Finite Center ◽  
Cartan Involution ◽  
Real Analytic

Let G be a connected real semisimple Lie group with finite center and θ be a Cartan involution of G. Suppose that K is the maximal compact subgroup of G corresponding to the Cartan involution θ. The coset space X = G/K is then a Riemannian symmetric space. In this paper, by choosing the reduced root system Σ0 = {α ∈ Σ | 2α /∈ Σ; α 2 ∈/ Σ} insteads of the restricted root system Σ and using the action of the Weyl group, firstly we construct a compact real analytic manifold Xb 0 in which the Riemannian symmetric space G/K is realized as an open subset and that G acts analytically on it, then we consider the real analytic structure of Xb 0 induced from the real analytic srtucture of AbIR, the compactification of the corresponding vectorial part.

scholarly journals Maximal convergence groups and rank one symmetric spaces

2007 ◽  
Vol 83 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Ara Basmajian ◽  
Mahmoud Zeinalian
Keyword(s):  
Symmetric Space ◽  
Hyperbolic Plane ◽  
Symmetric Spaces ◽  
Convergence Property ◽  
Noncompact Type ◽  
Hyperbolic Case ◽  
The Real ◽  
Convergence Group ◽  
Rank One ◽  
Möbius Groups

abstractWe show that the group of conformal homeomorphisms of the boundary of a rank one symmetric space (except the hyperbolic plane) of noncompact type acts as a maximal convergence group. Moreover, we show that any family of uniformly quasiconformal homeomorphisms has the convergence property. Our theorems generalize results of Gehring and Martin in the real hyperbolic case for Möbius groups. As a consequence, this shows that the maximal convergence subgroups of the group of self homeomorphisms of the d–sphere are not unique up to conjugacy. Finally, we discuss some implications of maximality.

Minimum and Conjugate Points in Symmetric Spaces

1962 ◽  
Vol 14 ◽  
pp. 320-328 ◽  
Author(s):  
Richard Crittenden
Keyword(s):  
Symmetric Space ◽  
Algebraic Structure ◽  
Tangent Bundle ◽  
Tangent Space ◽  
Symmetric Spaces ◽  
Conjugate Points ◽  
Minimum Cut ◽  
Cut Locus ◽  
Connected Case ◽  
Simply Connected

The purpose of this paper is to discuss conjugate points in symmetric spaces. Although the results are neither surprising nor altogether unknown, the author does not know of their explicit occurrence in the literature.Briefly, conjugate points in the tangent bundle to the tangent space at a point of a symmetric space are characterized in terms of the algebraic structure of the symmetric space. It is then shown that in the simply connected case the first conjugate locus coincides with the minimum (cut) locus. The interest in this last fact lies in its identification of a more or less locally and analytically defined set with one which includes all the topological interest of the space.

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