Questions tagged [symmetric-spaces]

A symmetric space is a differentiable manifold with the additional structure of a pseudo-Riemannian metric and which has many isometries.

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Rank of locally symmetric spaces in terms of flat immersions

Let $M$ be a complete locally symmetric space of finite volume and noncompact type. The rank of $M$ is usually defined as the rank of the symmetric space $\tilde{M}$ universally covering $M$, that is, ...
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Is the homology of the based loop space of a compact globally symmetric space a polynomial ring?

Let $X$ be a space. Then the homology group $H_*(\Omega X;\mathbb{Q})$ of the based loop space of $X$ is a $\mathbb{Q}$-algebra with the Pontryagin product given by loop concatenation. When $X=G$ is ...
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Dominant Weyl chamber and co-adjoint orbit for Riemann symmetric pair

Let $K$ be a compact Lie group with Lie algebra $\mathfrak{k}_0$. Suppose that $\sigma$ is an involutive automorphism of $K$ which defines a symmetric pair $(\mathfrak{k}_0,\mathfrak{k}_0^\sigma)$. ...
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Isometry group of Riemannian symmetric spaces and Lie groups

How do I prove the statement that for a Riemannian symmetric space $M$, the isometry group $Iso(M)$ is a Lie group? What can we say about the dimension of $Iso(M)$? And is its action transitive on $M$?...
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Relationship between distances on homogeneous spaces and their Lie groups

Consider the (round) sphere $M=\mathbb{S}^{n-1}$ as a homogeneous $O(n)$-space. Then for $x,y\in\mathbb{S}^{n-1}$ there is $g\in O(n)$ such that $y=g\cdot x$. Denote the Riemannian distance on $\...
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Coset decomposition of a Lie algebra with a compact symmetric subalgebra

If the Lie algebra $\mathbf G$ is connected and $\mathbf G = \mathbf K \oplus \mathbf P$ where $\mathbf K$ is a compact symmetric subalgebra, $$ [\mathbf K, \mathbf K]\subset \mathbf K, \ [\mathbf ...
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Embedding of Riemannian symmetric spaces $E_I$ and $E_{IV}$ into $E_6$ Lie group

In answer and comments to this mathoverflow question we have discussed possiblity of embedding Riemmanian symmetric spaces $E_I, E_{II}, E_{III},E_{IV}$ of dimension $42,40,32,26$ respectively into $...
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Ricci flat symmetric spaces are flat?

It is stated in Joyce's 'Compact Manifolds with Special Holonomy' (P.124) that if $M$ is a compact Riemannian symmetric space, then $M$ is Ricci flat implies $M$ is flat. I am having trouble seeing ...
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Addition property of Laplace-Beltrami eigenfunctions in symmetric spaces

Consider the eigenvalue equation for the Laplace-Beltrami operator on a manifold with metric $ds^2=|K|^{-1}[d\chi^2+\sin_K^2\chi(d\theta^2+\sin^2\theta\,d\phi^2)]$, where: $$\sin_K\chi=\left. \begin{...
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geodesic boundary of symmetric spaces

I want to understand the geodesic boundary of $X_3=G_3/K_3=SL_3(\mathbb{R})/SO_3(\mathbb{R})$. Apparently there three types of boundary points. So, my question is: Can someone give a description of ...
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Is an anti-symmetric relation “the biunivocal correspondence that associates to each point A the point A its symmetrical, and vice versa”?

The biunivocal correspondence that associates to each point A the point A 'its symmetrical, and vice versa, is called central symmetry of center O I need to take a step back to re-think pre-symmetry ...
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references for the quotient of a product of hyperbolic 2- and 3- spaces by $SL_2$ over a number ring

Let $F$ be a number field of degree $r+2s$ with ring of integers $\mathcal{O}_F$, $G=SL_2(F\otimes_{\mathbb{Q}}\mathbb{R})\cong SL_2(\mathbb{R})^r\times SL_2(\mathbb{C})^s$, $K=SO_2(\mathbb{R})^r\...
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What are the simply-connected non-compact irreducible symmetric spaces?

Would someone be able to list (or provide a reference to) the simply-connected non-compact irreducible symmetric spaces of rank $\ge 1$(as quotients of Lie groups $G/H$)? Any help would be ...
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Margulis' Arithmeticity theorem and arithmetic manifolds / locally symmetric spaces

I admit that there are several different definitions of an arithmetic manifold in the literature, but consider the following one: An arithmetic manifold is a quotient $M = \Gamma \backslash \mathbf{...
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If $H$ acts transitively on $G/K$, does it contain a copy of $AN$?

This is the question that I actually meant to ask in Which groups $H$ act transitively on a noncompact symmetric space $G/K$? I got confused about the definition of parabolic subgroups, so the answer ...
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Which groups $H$ act transitively on a noncompact symmetric space $G/K$?

All Lie groups here are assumed to be real. Let $M=G/K$ be a symmetric space of noncompact type and $H \subset G$. $H$ acts on $G/K$ by left-multiplication. If $H$ is a parabolic subgroup, then $H$ ...
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What do radially symmetric functions on a Riemannian symmetric space look like?

Let $M$ be a Riemannian manifold with isometry group $G$. We call a smooth function (on $M$, or on an appropriate neighborhood of $x_0$) radially symmetric about $x_0 \in M$ if it is invariant under ...