Questions tagged [symmetric-spaces]
A symmetric space is a connected Riemannian manifold in which at every point there exists a global self-isometry whose differential at the given point is minus identity.
199 questions
5
votes
0
answers
146
views
Almost complex manifolds with a $\text{U}(1)$ action fixing any point
Suppose we have a connected compact almost complex manifold $M$ such that for each point $p$ of $M$ there is an action $\phi$ of $\text{U}(1)$ on $M$ as diffeomorphisms preserving the almost complex ...
- 24.7k
3
votes
0
answers
95
views
Adjoint space of a compact irreducible symmetric space
From Helgason's book [1], Ch. VII, $\S10$, page 326-327:
Let $(\mathfrak u,\theta)$ be an orthogonal symmetric Lie algebra of the compact type and suppose $\mathfrak k_0$, the fixed point set of $\...
- 2,209
11
votes
0
answers
321
views
How close is the "bioctonionic projective plane" to an actual projective plane?
Boris Rosenfeld introduced some "generalized projective planes", Riemannian manifolds he called $(\mathbb{C} \otimes \mathbb{O})\text{P}^2, (\mathbb{H} \otimes \mathbb{O})\text{P}^2$ and $(\...
- 24.7k
0
votes
0
answers
107
views
Determining if a set intersects the orbit of another in Siegel upper half space
Consider Siegel upper half space, consisting of symmetric matrices $X+iY$ such that $Y$ is positive definite. This has an action of $\operatorname{Sp}_{2n}(\mathbb{Z})$ on it by generalized Möbius ...
- 29
2
votes
1
answer
156
views
Determining if a set is contained in Siegel's fundamental domain
Consider the Siegel upper half plane, i.e. symmetric complex matrices X + iY with positive definite imaginary part. This has an action of Sp_{2n}Z on it by generalized Möbius transformations.
There is ...
- 29
2
votes
0
answers
66
views
What is the derivative of a Killing field-valued function?
Let $M$ be a symmetric space and let $\mathfrak g$ be its Lie algebra. We have a natural bundle homomorphism
$$\alpha: M \times \mathfrak g \to TM.$$
Specifically, given $\xi \in \mathfrak g$, we can ...
- 1,280
3
votes
1
answer
231
views
Finding a paper by Takeuchi
I’m working on a project that involves compact symmetric spaces and I wanted to check a paper listed as a reference. I couldn’t find a paper copy of the journal in my school library. Hathitrust does ...
3
votes
0
answers
77
views
Levi-connection of special orthogonal as a symmetric space
Let SO(n) be the special orthogonal group with bi-invariant metric, $g(X, Y) = \frac{1}{2}trace(AB^{t})$. The Levi-Civita connection on SO(n) as a Lie group with bi-invariant metric is $\nabla_{X} Y = ...
- 83
3
votes
0
answers
115
views
Cusps of rank-one locally symmetric spaces
I've found in the literature these facts:
Any closed flat manifold is virtually (i.e. finitely covered by) a torus, and any finite-volume real hyperbolic manifold has virtually (i.e. is finitely ...
- 81
2
votes
0
answers
142
views
Normalizer of holonomy of symmetric spaces
Let $(M, g) = G/K$ be a symmetric space, i.e.
$G$ is a connected Lie group,
$\sigma \colon G \to G$ is an involution, with fixed locus $G^\sigma$,
$\mathfrak g = \mathfrak k \oplus \mathfrak m$ is ...
- 1,177
4
votes
1
answer
303
views
Is $\mathbb H^{p,q} _\mathbb C$ a symmetric space?
I asked a similar question yesterday and got negative votes. I was thinking I should ask the question in more details. I am very new to symmetric space stuff and if anyone knows this, please advise....
- 185
3
votes
1
answer
198
views
Siegel domain associated to a symmetric space
Consider the symmetric space
$$
\frac{\mathrm{SU}(2,1)}{\mathrm{S(U}(2) \times \mathrm{U}(1))}= \mathbb{H}_\mathbb C^2.
$$
For this symmetric space there is a Siegel domain (See the page 22 of Complex ...
- 185
1
vote
0
answers
63
views
Factor $M_i$ of affine symmetric product manifold $M_1 \times M_2$ is also affine symmetric?
Let $M_1$ and $M_2$ be smooth manifolds with linear connections $\mathcal{H}_i$. Suppose that $M_1 \times M_2$ with the product connection $\mathcal{H}_1 \times \mathcal{H}_2$ is an affine symmetric ...
- 810
5
votes
0
answers
170
views
Classifications of noncompact real semisimple Lie groups of rank $1$
Is there any classifications of noncompact real semisimple Lie groups of rank $1$? I know a few of above said groups, e.g. $SO_0(n,1), SU(n,1), Sp(n,1)$ and these are isometry groups of real, complex, ...
- 185
2
votes
0
answers
138
views
Space of singular geodesics in symmetric spaces of higher rank
It is known that for a large class of smooth manifolds $M$ (e.g. Euclidean space, hyperbolic space, the sphere, real, complex, quaternionic spaces, the Cayley plane, complex hyperbolic space) the ...
- 435
1
vote
0
answers
111
views
Fundamental group of the locally symmetric space $\Gamma\backslash G/K$
Let $G$ be a real Lie group, let $O$ be a maximal compact, let $A$ be a split torus and $K=OA$. Then $D=G/K$ is a symmetric space. Let $\Gamma$ be a neat arithmetic subgroup of $G$.
What can we say ...
user546695
2
votes
0
answers
93
views
Triangles in SL_d/SO_d
I was playing around with $X_d$ be the symmetric space of $SL_d (\mathbb{R})$, and found the following (vague) question:
Given $p,q,r \in X_d$, under which conditions $p,q,r$ lie in a totally ...
- 415
5
votes
1
answer
387
views
Central isogeny, Shimura varieties and exceptional cases
For a simple complex Lie algebra $\mathfrak g$, its weight lattice is not equal to the root lattice (i.e. the center of its simply connected form is a non-trivial finite group) iff $\mathfrak g$ is of ...
- 7,442
1
vote
0
answers
87
views
Analytic and Algebraic Brauer groups of complex symmetric spaces
Let $G$ be a connected reductive group and $K=G^{\sigma}$ where $\sigma$ is an involution of $G$. Could someone please help me to understand what will be the (Analytic) Algebraic Brauer groups of the ...
- 39
4
votes
1
answer
260
views
Compact symmetric spaces and sub-root systems
Given two semisimple complex Lie algebras $\frak{g}$ and $\frak{n}$ such that the root system of $\frak{n}$ arises as a sub-root system of the root system of $\frak{g}$, does this then imply that $\...
3
votes
1
answer
156
views
Representation of Lie groups inducing a quasi-isometric embedding of their symmetric spaces
Let $G_{1}$ and $G_{2}$ be connected semisimple real Lie groups with no compact factors and finite center and let $K_{1}$ and $K_{2}$ denote some fixed choice of their maximal compact subgroups, ...
2
votes
1
answer
176
views
Why automorphism group of a Hermitian symmetric domain has trivial center?
Definition: A Hermitian symmetric domain is a Hermitian manifold that is connected, homogeneous, has a symmetry at some point (by homogenity hence every point), and has negative curvature.
I want to ...
- 1,051
2
votes
0
answers
83
views
De Rham product decomposition theorem in a particular setting
Let $G$ be a Lie group, and $H$ a Lie subgroup of $G$ such that $G/H\sim \mathcal M$ is a homogeneous space diffeomorphic to $\mathbb{R}^n$, equipped with an invariant Riemannian metric $g$. If this ...
- 401
4
votes
3
answers
390
views
References for $K$-orbits in $G/B$
Let $G$ be a reductive group, $K$ a symmetric subgroup of $G$ (e.g., fixed point of an involution), and $B$ a Borel subgroup of $G$. Then it is well known that $G/B$ has finitely many $K$-orbits. ...
- 751
5
votes
0
answers
110
views
Spherical functions in the space of functions on real Grassmannians
Let $G=O(n)$ be the orthogonal group. Let $K=S(O(k)\times O(n-k))$ be the subgroup of $O(n)$.
Then the pair $(G,K)$ is symmetric, and the homomogeneous space $G/K$ is the Grassmannian of $k$-...
- 23.1k
1
vote
0
answers
142
views
Harmonic analysis of vector bundles on symmetric spaces
$\DeclareMathOperator{\SO}{SO}$This is a follow-up to my previous question.
Given a semisimple symmetric space $M\simeq G/H$, in particular, the real hyperbolic space $H_{p,q}\simeq \SO(p,q)/\SO(p,q-1)...
- 164
2
votes
0
answers
205
views
A property of the Riemannian metric on the symmetric space $X_\textsf{d}=\textsf{SL}(d,\Bbb R)/\textsf{SO}(d)$
Consider the symmetric space $X_\textsf{d}=\textsf{SL}(d,\Bbb R)/\textsf{SO}(d)$. For any $M\in X_d$, by transporting the Killing form on $T_I(X_d)$ (the space of symmetric matrices with trace zero) ...
- 1,201
4
votes
0
answers
190
views
Relation between two Harish-Chandra homomorphisms
Let $\mathfrak{g}$ be a Lie algebra admitting a triangular decomposition $\frak g = N^-\oplus h\oplus N^+$ and $\gamma$ the classic Harish-Chandra isomorphism defined on the center $\frak Z$ of the ...
- 141
2
votes
1
answer
625
views
How to prove Siegel upper half plane is a hermitian symmetric space
There is a statement that is Siegel upper half plane of genus g, $\mathbb{H}_g:=\left\{Z=X+i Y \in M_n(\mathbb{C}) \mid X, Y \text { real }, Z=Z^{T}, Y=\operatorname{Im} Z>0\right)$ is isomorphic ...
- 21
1
vote
2
answers
263
views
Classification of compact isotropy irreducible homogeneous Kaehler manifolds
Is classification of compact isotropy irreducible homogeneous Kaehler(-Einstein) manifolds known?
Here, a homogeneous space is called isotropy irreducible if the isotropy representation is irreducible....
- 21
1
vote
0
answers
51
views
Bi-$M$-invariant measure on a Riemannian symmetric space
Let $G$ be a noncompact connected semi simple Lie group. Let $K$ be a maximal compact subgroup and $G=K\overline{A_{+}}K$ be a Cartan decomposition of $G.$ Let $M=Z_{K}(\mathfrak{a})$. Then how to ...
- 1,980
8
votes
2
answers
720
views
Spherical roots, restricted roots, and the dual group of a symmetric variety
Let $k$ be an algebraically closed field of characteristic $0$ and $G$ a semisimple simply-connected group over $k$. Consider a symmetric variety of the form $X=G/H$, for $H=G^\theta$ the fixed point ...
- 553
1
vote
0
answers
51
views
Hermitian locally symmetric space with nonnegative bisectional curvature
Let $(M,g)$ be an Hermitian locally symmetric space with nonnegative bisectional curvature. Suppose the fundamental group of $M$ is finite, can we prove that $M$ is simply-connected?
- 901
9
votes
2
answers
551
views
Almgren's regularity Theorem ; a simple example?
Let me remind Almgren's regularity Theorem: the singular set of area-minimizing surface has codimension at least $2$. I wish to share here a simple example in low dimension, although I don't know ...
- 53.1k
5
votes
1
answer
506
views
Buildings as generalizations of symmetric spaces
In almost every introductory notes on Tits buildings these are motivated
as structures capturing/ sharing several features of symmetric spaces. Could somebody elaborate what are precisely the main ...
- 3,878
1
vote
0
answers
141
views
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 ...
- 401
5
votes
1
answer
977
views
Definition of locally symmetric space of reductive groups
This might seems like a bit of philosophical question and so maybe if I keep reading a bit more, I might get my answer. But, I ask nonetheless.
In my attempt to study Shimura varieties, I came across ...
- 939
4
votes
0
answers
364
views
The convention of Fourier transform on symmetric spaces
When trying to understand the Plancherel formula of reductive symmetric space of Harish-Chandra class, I get confused on the convention of Fourier and related transforms.
$\newcommand{\H}{\mathcal{H}}
...
- 164
3
votes
1
answer
351
views
Cut locus for simply connected manifolds
Let $M$ be a connected and simply connected compact Riemannian manifold (without boundaries). Fix a point $p\in M$.
The diameter set $D_p$ of $p$ is the set of points that maximize the distance from $...
- 809
2
votes
0
answers
68
views
Multiplicative invariants of non-reduced root systems
It is a well known fact (cf. [1] VI.3.4 Thm. 1) that if $\Phi$ is a (reduced) root system with weight lattice $P$ and $W$ is the Weyl group of this root system, then the algebra of invariant ...
- 553
1
vote
0
answers
175
views
A question about Araki's 1962 paper on classification of irreducible symmetric spaces
I am looking at Sôhô Araki's 1962 paper for the classification of real semisimple lie algebras. Here's the link to the paper: On root systems and an infinitesimal classification of irreducible ...
- 419
5
votes
1
answer
215
views
Maximum symmetry metric on Cayley plane $ F_4/{\operatorname{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; see ...
- 4,577
11
votes
2
answers
1k
views
Is there a contractible hyperbolic 3-orbifold of finite volume?
Let $\mathbb{H}^3:=\operatorname{SO}(3,1)/\operatorname{O(3)}$.
Is there a lattice $\Gamma$ in $\operatorname{SO}(3,1)$ such that
\begin{equation}
X:=\mathbb{H}^3/\Gamma
\end{equation}
is contractible?...
- 423
1
vote
0
answers
78
views
A generalised spectral theorem for symmetric *spaces* generalising the one for symmetric matrices
The set of $n \times n$ symmetric matrices over $\mathbb R$ form a symmetric space. The relevant Lie group is $G = GL_n(\mathbb R)$ and the relevant involution is $\sigma(X)=X^{-T}$; it follows then ...
- 5,023
6
votes
0
answers
193
views
Maximum symmetry metric on irreducible compact symmetric space
Let $ M $ be a compact connected manifold. The degree of symmetry of $ M $, denoted $ N(M) $, is the maximum of the dimensions of the isometry groups of all possible Riemannian structures on $ M $. ...
- 4,577
5
votes
1
answer
282
views
Can all hermitian symmetric spaces be realised as coadjoint orbits?
Here is what I know. Assume $M\cong G/K$ is an irreducible hermitian symmetric space. Denote the Lie-algebra of $K$ by $\mathfrak{t}$. Proposition 1.2. chapter 3 in
Wienhard - Bounded cohomology and ...
- 61
13
votes
0
answers
300
views
Symmetric spaces are quandles. Is this important?
For concreteness, let's consider a connected reductive Lie group $G$, and an involution $\theta$ on it. Then the associated symmetric space $X=G/G^\theta$ has the structure of an involutive quandle: ...
- 2,621
3
votes
0
answers
168
views
Conjugacy classes of Cartan subspaces in parahermitian symmetric spaces
Are there any good tables of the numbers of conjugacy classes of Cartan subspaces in pseudo-Riemannian symmetric spaces? Or a good method to count them? In particular, I am interested in the ...
- 1,004
2
votes
2
answers
429
views
Are the automorphism groups of simple symmetric cones algebraic groups?
This question arises when I tried to understand Chapter 2 of the celebrated book "Smooth compactification of locally symmetric varieties" by Ash–Mumford–Rapoport–Tai.
The setting is as ...
- 339
11
votes
1
answer
958
views
Explicit construction of a (the?) dual symmetric space
I am looking for a reference, proof or disproof of the fact that every Riemannian globally symmetric space of compact (non-compact) type has a "dual", which is of non-compact (compact) type.
...
- 133