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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Fixed point of the isotropy group in a symmetric one

Let $V$ be Eucldiean vector space of finite dimension. Let $C$ a symetric cone, that is to say: open convex self-dual homogeneous: $\forall x,y\in C, \exists M\in G, Mx=y$ where $G=\lbrace M| M\in GL(...
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Equivalence between bi-invariant metrics on Lie groups and Symmetric spaces

Let $G$ be a simply connected Lie group with Lie algebra $\mathfrak{g}$ and $K$ a connected closed Lie subgroup of $G$ with Lie algebra $\mathfrak{s}$. Then $G/K$ is a homogeneous space. Equip $G$ ...
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Do there exist theorems of de Sitter geometry, just as there are theorems of Minkowski and anti-de Sitter geometry?

There are theorems of Euclidean, hyperbolic, elliptic and Minkowski geometry. I'm wondering about planar de Sitter geometry. Regarding planar anti-de Sitter geometry, based on my understanding of ...
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Question about decomposition about splitting of symmetirc spaces of compact type

I get stuck in the following question: Why does a locally symmetric space of compact type $M$ split locally irreducible components of dimension $\geq 2$ which are Einstein? In particular, why are all ...
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Why is $G/K$ a space of non-positive curvature, where $G$ is a complex reductive group and $K$ is its maximal compact subgroup

I am trying to understand the proof of Kempf-Ness theorem, which relates polystability (Mumford's definition) and zeroes of moment maps. A key part of this proof is the following idea, that for a ...
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Let V a real vector space of finite dimension, with a positive definite inner product and let A a symmetric operator in this space

I want to know how can I generalize this excersice. Let V a real vector space of finite dimension, with a positive definite inner product and let A a symmetric operator in this space a)We have $g(v,w)...
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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 ...
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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 ...
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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 ...
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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 $SL(n) / SO(n)$? ...
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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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Proof of $\mathsf{SU}(3)/T^2$ is not a symmetric space

How to see that $\mathsf{SU}(3)/T^2$ is not a symmetric space? This is not obvious to me. How to see that it admits a metric of positive curvature? The only clue that I know is the submersion $\...
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About the proof of Proposition 6.45 on Ziller's notes

I'm currently going through W. Ziller's notes on symmetric spaces, and I've come across one argument he makes which I can't seem to wrap my head around. Suppose $(G,K)$ is a symmetric pair of the ...
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Exponentiating two elements in lie group

This must be a silly question, but while reading the book of Borel and Ji on compactifications of symmetric and locally symmetric spaces I got completely stuck on the notation. Does the notation $x^y$ ...
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Is the Ricci tensor of a symmetric space proportional to the killing form?

In Besse's book on Einstein manifolds, one can read the following theorem 7.73 Theorem: The Ricci curvature of a Riemannian symmetric space satisfies: $$r=-\frac{1}{2}B_{\vert\mathfrak p}$$ Here $r$ ...
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Local obstruction to having a complete extension

This question arose while I was reading Helgason's book on symmetric spaces. In chapter IV section 5, one can read the following: Let $M$ be a Riemannian manifold, $p$ a point in $M$. In general it ...
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Riemannian exponential map on symmetric spaces of non-compact type

Let $S$ be a symmetric space of non-compact type, specifically let $S= SL(n)/SO(n)$ for some $n\geq 2$. I know that such spaces are geodesically complete, simply connected, and have non-positive ...
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Curvature invariant under isometry but not parallel transport?

I have read that the curvature is invariant under isometry in the general, though I'm urrently reading up a little on symmetric spaces and I've seen that we have $\nabla R=0$ iff the space is locally ...
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Representations of $\text{Sp}(n,1)$ on $\mathbb{R}^d$

Can anyone help me understand why there are no nontrivial homomorphisms $\text{Sp}(n,1) \to \text{GL}(d,\mathbb{R})$ if $n$ is sufficiently large relative to $d$? By definition, $\text{Sp}(n,1)$ is ...
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Involutive automorphism of simple Lie algebra

Let $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ be a Cartan decomposition for a noncompact real simple Lie algebra $\mathfrak{g}$ corresponding to a Cartan involution $\theta$, where $\mathfrak{k}$ is ...
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Examples: (Symmetric) Quotients of Euclidean Space

What are some examples of $n$-dimensional topological manifolds $M$ (possibly with boundary), which can be written as a quotient $\mathbb{R}^d/G$ for some topological group acting on $\mathbb{R}^d$?
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Looking for reference: If a Riemanian manifold is foliated by max symmetric submanifolds, then coordinates can always be chosen such that ...

In Weinberg's book on General Relativity in section 13.5 it is shown that, loosely stated, if a Riemannian manifold $(M,g)$ of dim $m$ is composed of maximally symmetric submanifolds $(N,h)$ of dim $n$...
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Space form structure for non orientable surfaces

The projective plane is homogeneous $$ N_0=P^2 \cong SO_3/O_2 $$ The connected sum of two projective planes is the Klein bottle $ K= N_1 $ which is also homogeneous (for the construction of this ...
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Why is the bilinear form $B_{\theta}$ (defined via Killing form) positive definite?

Let $\frak{g}$ be a matrix Lie algebra with Cartan involution $\theta : X\mapsto -X^*$ (negative conjugate transpose). Let $B$ be the bilinear symmetric Killing form $B(X,Y)=\operatorname{tr}(\...
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2 votes
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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 ...
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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 $(\mathfrak{k}_0,\mathfrak{k}_0^\sigma)$. ...
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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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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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2 votes
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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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6 votes
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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 ...
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2 votes
1 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 ...
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