A symmetric space is a differentiable manifold with the additional structure of a pseudo-Riemannian metric and which has many isometries.
Questions tagged [symmetric-spaces]
71 questions
7
votes
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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 $\mathsf{SU}(3)\to \mathsf{SU}(3)/T^2$ if I am not…
C.F.G
- 8,789
7
votes
1 answer
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 is impossible to find any neighborhood $N$ of $p$…
Adam Chalumeau
- 3,333
6
votes
1 answer
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 appreciated!
Sergio Charles
- 1,396
5
votes
1 answer
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 covariantly constant on locally symmetric spaces, and so these are locally…
Selene
- 1,123
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5
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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, it is the maximal dimension of a flat totally…
abenthy
- 963
5
votes
1 answer
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 $\mathbb{S}^{n-1}$ by $d_{\mathbb{S}^{n-1}}$.…
Sven Pistre
- 1,390
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4
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The symmetric space $\operatorname{SO}(n,\mathbb C) / \operatorname{SO}(n)$
I am looking for a reference for the symmetric space $\operatorname{SO}(n,\mathbb C) / \operatorname{SO}(n)$; I haven't been able to find any references on about it online. In particular, I would like to know a "geometric interpretation" of the…
Joseph Kwong
- 101
4
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0 answers
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 exact sequence of the form
$$ \cdots \to…
Hyeongmuk LIM
- 846
4
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0 answers
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…
Ian Gershon Teixeira
- 10,176
4
votes
1 answer
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 of $\mathfrak{g}$ such that $d_e \sigma…
Diego Artacho
- 549
4
votes
0 answers
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-symmetric non-compact quaternion-Kähler spaces (of…
AMA
- 41
4
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2 answers
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 noncompact type with Cartan decomposition…
Johnny El Curvas
- 1,722
4
votes
1 answer
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 a compact simply-connected Lie group, we know that…
ChiHong Chow
- 201
4
votes
1 answer
Parameterization Of Grassmannian
Is there an explicit formula to parameterize the Grassmannian $P$ (which is a $(a+b)\times (a+b)$ dimensional matrix)
$$P\in \frac{U(a+b)}{U(a)\times U(b)}$$
by $a\times b$ complex independent parameters?
(One may normalize $P$ such that…
user34104
- 573
4
votes
0 answers
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 to that question is something that I'm actually…
user538044
- 523