# 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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### Restriction of a Lie group automorphism to the subgroup associated to an invariant Lie subalgebra

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$, and let $\sigma \colon G \rightarrow G$ be a Lie group automorphism such that $\sigma^2 = \text{id}_G$. Let $\mathfrak{h}$ be a Lie subalgebra ...
1 vote
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### Explanation of the relationship between geometric and algebraic qualities of Globally Riemannian Symmetric Spaces.

If you google search Globally Riemannian Symmetric Spaces, you will receive several links which provide a geometric understanding of these spaces. I mean specifically you will get stuff about ...
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### $(G,K)$ is a Gelfand pair iff $C_c(G||K)$ is commutative

In my definition, a pair $(G, K)$, of a group $G$ and a compact subgroup $K$ of $G$, is said to be a Gelfand pair if the subalgebra $L^1(G||K)=\{f\in L^1(G): f^{\#}=f \}$ of $K$-bi invariant $L^1$ ...
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### Why does the Lehmer's conjecture imply the Short Geodesic Conjecture?

I need some translation help. In this article (https://homeweb.unifr.ch/kellerha/pub/IML-2013summer4-01.pdf), on page 15 of the pdf, it says that the "short geodesic conjecture" is a ...
1 vote
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### $\pi\circ \exp|_\mathfrak{m}$ surjective on noncompact homogeneous space

Let $(G,\langle\,,\,\rangle)$ a connected Lie group with bi-invariant semi-Riemannian metric. Let $H$ be a closed subgroup, and denote by $h$ the left-invariant metric on the homogeneous space $N=G/H$ ...
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### Classification of Symmetric Spaces of Dimension $n$ and Rank $k$?

(Rather than edit, I'm making a new post.) In Classification of Closed, Locally Symmetric Spaces of Dimension $n$ and Rank $k$?, I asked about closed, locally symmetric spaces. I now realize what I ...
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### Classification of Closed, Locally Symmetric Spaces of Dimension $n$ and Rank $k$?

Let $M^n$ be a closed locally symmetric space. Then $M$ a Riemannian manifold, is compact, has empty boundary, and has $\nabla R = 0$, where $R$ is the curvature tensor. If $M$ is a closed locally ...
1 vote
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### Volume of a geodesic ball in $SL(n) / SO(n)$?

Crossposted on MO: https://mathoverflow.net/questions/404944/volume-of-a-geodesic-ball-in-sln-son Question: What is the volume of a geodesic ball of radius $r$ in the symmetric space $SL(n) / SO(n)$? ...
1 vote
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### Isometries of the symmetric space of $PSL_d(\mathbb{R})$

Let $G = PSL_d(\mathbb{R})$ and $K = SO(d)$. It is a known fact that the symmetric space $G/K$ can be identified with the space $X_d$ defined by the space of inner products in $\mathbb{R}^d$ up to ...
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### What are locally parallelizable manifolds?

I came across this concept on this wiki page regarding killing vector field. The last sentence in section "Cartan Involution" says that "Equivalently, the curvature tensor is ...
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### Which non-compact quaternion-Kähler spaces are Kähler?

The list of quaternion-Kähler compact symmetric spaces can be found here. I am curious to know which of the non-compact versions of these spaces are Kähler. If the answer is known also for non-...
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### Lie group involutive automorphism inducing a symmetric space structure on the quotient by its fixed-point-set

I'm trying to understand one example of a symmetric space in Postnikov's Riemannian Geometry book but I'm unsure if I'm messing up one of the identifications, or just the algebra. Here's the setup: A ...
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### Is Complex tori formal?

A manifold is called geometrically formal when all wedge products of harmonic forms are harmonic. As it says here: https://arxiv.org/pdf/math/0004009.pdf: On a general Riemannian manifold, wedge ...
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### Qustions on the orbits of weyl group and group actions

I am an undergraduate in physics and know little about math. I know about some basic ideas of Lie groups and Lie algebras like roots, weyl group, weyl chambers but I am ignorant about complexification,...
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### Locally symmetric spaces have parallel Riemannian curvature

I'm trying to prove the following result: If $(M,g)$ is a locally symmetric Riemannian manifold, then the Riemannian curvature tensor is parallel: $\nabla Rm \equiv 0$. By "locally symmetric", I ...
1 vote
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### Isometry group of $3$-sphere

I would like to know how I can prove that $SU(2)$ acts transitively on $S^{3}$. Currently, I want to show that $SU(2)$ is a group of isometries of $S^{3}$.
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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 $(\mathfrak{k}_0,\mathfrak{k}_0^\sigma)$. ...
1 vote
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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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### 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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### 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 ...
### 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 $H$ ...
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 ...