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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(locally) symmetric spaces where every conformal transformation is an isometry
By an argument using Liouville's theorem for conformal maps Conformal automorphism of $H^n$ it can be shown that every conformal automorphism of a hyperbolic manifold is an isometry.
Are there any ...
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Some confusion about Riemannian symmetric space
Recently I'm learning some basic theories about symmetric spaces from "Differential geometry, Lie groups, and symmetric space" written by Sigurdur Helgason, and I have some confusion. I Hope ...
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Computation of the LES of homotopy groups associated with compact symmetric spaces
I am looking for an efficient way to compute the homotopy groups, as well as morphisms between them, of certain matrix groups and compact symmetric spaces. To be specific, I want to determine the long ...
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How to describe all metrics invariant by $O(3)$ on a given space?
I want to describe all pseudo-Riemannian metrics (symmetric covariant non degenerate 2-tensor fields) invariant by $O(3)$ on different manifolds ($\mathbb R^3$, the sphere $S^2$, and $]-\epsilon,\...
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Need a help to find a reference or books to noncommutative symmetric spaces [closed]
I have a diploma work on a "Integration on noncommutative symmetric spaces". However, I cannot find any references to them. It would be great if you can give some advices about what to read ...
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Symmetric cones and symmetric spaces
I start by stating what I think I understood on symmetric cones (https://en.wikipedia.org/wiki/Symmetric_cone). Let $\mathcal{C}$ be a symmetric cone in a vector space $V$. There are Riemannian ...
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Question on Riemannian manifold with maximally symmetric submanifold
It is proved in Weinberg's text (Gravitation and cosmology, ch.13 section 5) that if a manifold has a submanifold which is maximally symmetric (has maximal number of killing vector fields w.r.t. the ...
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curvature tensor of symmetric spaces
I'm trying to understand the following theorem about the curvature tensor of a symmetric space:
Let $R$ the curvature tensor of the space $G/K$ corresponding to the Riemannian structure $Q$ ,then at ...
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Does the QR/Iwasawa decomposition have any geometric meaning for $2 \times 2$ complex matrices?
It's well known that $2 \times 2$ complex matrices act on the complex plane (with a point at infinity) by $\begin{pmatrix}a & b \\ c & d \end{pmatrix}\cdot z = \frac{az + b}{cz + d}$.
Based on ...
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A doubt from Araki's 1962 paper on classification of irreducible symmetric spaces
I am looking at Soho Araki's 1962 paper for the classification of real semisimple lie algebras.Here's the link to the paper: Araki's paper.In page $9$, proposition $2.2$,there is a criteria for $\psi\...
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Proof that $ SO_n $ is maximal in $ SU_n $
$ SO_n $ is maximal among the proper connected closed subgroups of $ SU_n $ (at least this is true for small $ n $). What is a conceptual proof of this fact? Equivalently what is a proof that $ \...
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Is the natural metric on an irreducible compact symmetric space Einstein?
Let $ G/K $ be an irreducible compact symmetric space. Let $ g $ be the pushforward onto $ G/K $ of the biinvariant metric on $ G $. Is $ g $ an Einstein metric on $ G/K $?
This is true for $ S^n=SO_{...
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Symmetrization map over the polynomial ring of a vector space.
Let $V$ be a finite-dimensional complex vector space. Is the morphism
\begin{gather*}
\mathrm{Sym}^{\bullet}(V \oplus V^*) \to D(V) \cong \frac{\mathrm{T}^{\bullet}(V\oplus V^*)}{I} \,, \\[0.5em]
(v_1,...
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Maximum symmetry metric on Cayley Plane $ F_4/Spin(9) $
The maximum symmetry metric on real projective space $ \mathbb{RP}^n $ is the round metric.
The maximum symmetry metric on complex projective space $ \mathbb{CP}^n $ is the Fubini-Study metric.
https:/...
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Minimal spheres in rank one symmetric spaces
Let $M^n$ be a compact rank one symmetric space endowed with its canonical metric. For a given point $p \in M$, does there exist a positive number $r$ such that the sphere
$$S_r(p) = \{ x \in M: d(x,p)...
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Analogues of Iwasawa Decomposition for $\operatorname{SL}_2(\mathbb{R})/\operatorname{SO}(1,1)(\mathbb{R})$
Consider the group $\operatorname{SL}_2(\mathbb{R})$. The set of positive-definite symmetric $2 \times 2$ matrices of determinant $1$ constitutes a symmetric space $S^+$ for $\operatorname{SL}_2(\...
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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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$\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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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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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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