scholarly journals Equivariant harmonic maps depend real analytically on the representation

2021 ◽  
Author(s):  
Ivo Slegers
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Harmonic Maps ◽  
Main Tool ◽  
Target Space ◽  
Fixed Target ◽  
Compact Type ◽  
Proper Action ◽  
Real Analytic

AbstractWe consider harmonic maps into symmetric spaces of non-compact type that are equivariant for representations that induce a free and proper action on the symmetric space. We show that under suitable non-degeneracy conditions such equivariant harmonic maps depend in a real analytic fashion on the representation they are associated to. The main tool in the proof is the construction of a family of deformation maps which are used to transform equivariant harmonic maps into maps mapping into a fixed target space so that a real analytic version of the results in [4] can be applied.

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 Dynamics of the heat semigroup on symmetric spaces

2009 ◽  
Vol 30 (2) ◽  
pp. 457-468 ◽  
Author(s):  
LIZHEN JI ◽  
ANDREAS WEBER
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Chaotic Behavior ◽  
Heat Semigroup ◽  
Euclidean Spaces ◽  
Compact Type ◽  
Heat Semigroups

AbstractThe aim of this paper is to show that the dynamics of Lp heat semigroups (p>2) on a symmetric space of non-compact type is very different from the dynamics of the Lp heat semigroups if 1<p≤2. To see this, we show that certain shifts of the Lp heat semigroups have a chaotic behavior if p>2, and that such a behavior is not possible in the cases 1<p≤2. These results are compared with the corresponding situation for Euclidean spaces and symmetric spaces of compact type, where such a behavior is not possible.

scholarly journals Some results on Semisimple Symmetric Spaces and Invariant Differential Operators

2017 ◽  
Vol 116 (2) ◽  
Author(s):  
Trần Đạo Dõng
Keyword(s):  
Symmetric Space ◽  
Open Subset ◽  
Symmetric Spaces ◽  
Differential Operators ◽  
Analytic Manifold ◽  
Invariant Differential Operators ◽  
Semisimple Symmetric Space ◽  
Real Analytic Manifold ◽  
Invariant Differential ◽  
Real Analytic

<pre>Let X = G/H be a semisimple symmetric space of non-compact style. Our purpose is to construct a compact real analytic manifold in which the semisimple symmetric space X = G/H is realized as an open subset and that $G$ acts analytically on it.</pre><pre> By the <span>Cartan</span> decomposition <span>G = KAH,</span> we must <span>compacify</span> the <span>vectorial</span> part <span>A.$</span></pre><pre> In [6], by using the action of the Weyl group, we constructed a compact real analytic manifold in which the semisimple symmetric space G/H is realized as an open subset and that G acts analytically on it.</pre><pre>Our construction is a motivation of the <span>Oshima's</span> construction and it is similar to those in N. <span>Shimeno</span>, J. <span>Sekiguchi</span> for <span>semismple</span> symmetric spaces.</pre><pre>In this note, first we will <span>inllustrate</span> the construction via the case of <span>SL (n, </span>R)/SO_e (1, n-1) and then show that the system of invariant differential operators on X = G/H extends analytically on the corresponding compactification. </pre>

scholarly journals On splitting rank of non-compact-type symmetric spaces and bounded cohomology

2018 ◽  
Vol 12 (02) ◽  
pp. 465-489
Author(s):  
Shi Wang
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Maximal Dimension ◽  
Bounded Cohomology ◽  
Compact Type ◽  
Totally Geodesic Submanifold ◽  
Totally Geodesic ◽  
Geodesic Submanifold ◽  
Higher Rank

Let [Formula: see text] be a higher rank symmetric space of non-compact type where [Formula: see text]. We define the splitting rank of [Formula: see text], denoted by [Formula: see text], to be the maximal dimension of a totally geodesic submanifold [Formula: see text] which splits off an isometric [Formula: see text]-factor. We compute explicitly the splitting rank for each irreducible symmetric space. For an arbitrary (not necessarily irreducible) symmetric space, we show that the comparison map [Formula: see text] is surjective in degrees [Formula: see text], provided [Formula: see text] has no direct factors of [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. This generalizes the result of [J.-F. Lafont and S. Wang, Barycentric straightening and bounded cohomology, to appear in J. Eur. Math. Soc.] regarding Dupont’s problem.

scholarly journals Helgason spheres of compact symmetric spaces and immersions of finite type

2001 ◽  
Vol 63 (2) ◽  
pp. 243-255
Author(s):  
Bang-Yen Chen
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Finite Type ◽  
Symmetric Spaces ◽  
Unit Vector ◽  
Geodesic Sphere ◽  
Compact Type ◽  
Totally Geodesic ◽  
Compact Symmetric Space ◽  
Unit Speed

A unit speed curve γ = γ(s) in a Riemannian manifold N is called a circle if there exists a unit vector field Y(s) along γ and a positive constant k such that ∇sγ′(s) = kY(s), ∇sY(s) = −kγ′(s). A maximal totally geodesic sphere with maximal sectional curvature in a compact irreducible symmetric space M is called a Helgason sphere. A circle which lies in a Helgason sphere of a compact symmetric space is called a Helgason circle. In this article we establish some fundamental relationships between Helgason circles, Helgason spheres of irreducible symmetric spaces of compact type and the theory of immersions of finite type.

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).

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