# 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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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 $\... 0answers 29 views ### Coset decomposition of a Lie algebra with a compact symmetric subalgebra If the Lie algebra$\mathbf G$is connected and$\mathbf G = \mathbf K \oplus \mathbf P$where$\mathbf K$is a compact symmetric subalgebra, $$[\mathbf K, \mathbf K]\subset \mathbf K, \ [\mathbf ... 0answers 17 views ### Embedding of Riemannian symmetric spaces E_I and E_{IV} into E_6 Lie group In answer and comments to this mathoverflow question we have discussed possiblity of embedding Riemmanian symmetric spaces E_I, E_{II}, E_{III},E_{IV} of dimension 42,40,32,26 respectively into ... 1answer 50 views ### Ricci flat symmetric spaces are flat? It is stated in Joyce's 'Compact Manifolds with Special Holonomy' (P.124) that if M is a compact Riemannian symmetric space, then M is Ricci flat implies M is flat. I am having trouble seeing ... 0answers 48 views ### Addition property of Laplace-Beltrami eigenfunctions in symmetric spaces Consider the eigenvalue equation for the Laplace-Beltrami operator on a manifold with metric ds^2=|K|^{-1}[d\chi^2+\sin_K^2\chi(d\theta^2+\sin^2\theta\,d\phi^2)], where:$$\sin_K\chi=\left. \begin{... 0answers 11 views ### 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 ... 0answers 15 views ### 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 ... 1answer 40 views ### 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\...
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 ...