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.
182 questions
22
votes
2
answers
1k
views
Does $\mathrm{E}_7/(\mathrm{SU}_8/(\mathbb{Z}/2))$ carry an almost complex structure?
Recall the list of irreducible simply connected inner symmetric spaces of compact type in dimension $4k+2$:
Hermitian symmetric spaces (one can write them down explicitly);
Grassmannians of oriented ...
- 2,140
22
votes
2
answers
2k
views
"isotropic" subspaces of a simple Lie algebra
Let $\bf g$ be a finite-dimensional real simple Lie algebra of compact type and let $\left<-,-\right>$ denote the positive-definite inner product induced from the negative of the Killing form. ...
- 33.3k
21
votes
2
answers
1k
views
Geometric interpretation of exceptional symmetric spaces
Elie Cartan has classified all compact symmetric spaces admitting a compact simple Lie group as their group of motion.There are 7 infinite series and 12 exceptional cases. The exceptional cases are ...
- 3,022
18
votes
1
answer
1k
views
How come Cartan did not notice the close relationship between symmetric spaces and isoparametric hypersurfaces?
Elie Cartan made fundamental contributions to the theory of Lie groups and their geometrical applications. Among those, we can list the introduction of the remarkable family of Riemannian symmetric ...
- 4,729
15
votes
4
answers
2k
views
Applications of Jordan algebras
Jordan algebras are non-associative algebras satisfying a somewhat strange (to me) list of axioms, see wikipedia. Basic examples are real symmetric and complex hermitian matrices with the product $A\...
- 21.8k
15
votes
3
answers
1k
views
Is SO(2n+1)/U(n) a symmetric space?
I am a physics student with only a rudimentary knowledge of differential geometry, so please feel free to point out if I miss something elementary / trivial.
According to https://arxiv.org/abs/1408....
- 305
15
votes
2
answers
653
views
Invariant differential operators on real Grassmannians
I am looking for an explicit description of the algebra of $SO(n)$- or, better, $O(n)$-invariant differential operators on the real Grassmann manifolds of $k$-dimensional linear subspaces in the ...
- 21.8k
15
votes
0
answers
926
views
How does duality of symmetric spaces explain the hyperbolic cosine theorem?
There is a well-known duality between compact symmetric spaces and symmetric spaces of noncompact type. Basically it goes as follows: If $$G/K$$ is a symmetric space of noncompact type, $$g=k+p$$ the ...
- 10.4k
14
votes
2
answers
3k
views
Volume of fundamental domain and Haar measure
In my research, I do need to know the Haar measure. I have spent some time on this subject, understanding theoretical part of the Haar measure, i.e existence and uniqueness, Haar measure on quotient. ...
- 2,508
14
votes
1
answer
737
views
Algebraic characterization of the curvature operator of symmetric spaces
My question is the following :
Given an algebraic curvature operator $R\in S^2_B(\Lambda^2\mathbb{R}^n)$, is there an a simple criterion to know if this curvature operator can occur as the ...
- 4,123
12
votes
3
answers
2k
views
Why symmetric spaces?
In Bar-Natan's "Knots at Lunch" seminar at the University of Toronto, we are currently discussing a talk by Alekseev at Montpellier about Rouvière's expansion of the Duflo isomorphism to the ...
- 22.1k
12
votes
1
answer
338
views
Geodesic preserving diffeomorphisms of constant curvature spaces
Let $X$ be either Euclidean space $\mathbb{R}^n$, the sphere $\mathbb{S}^n$, or hyperbolic space $\mathbb{H}^n$.
I would like to have a classification of all diffeomorphisms $X\to X$ which map ...
- 21.8k
12
votes
1
answer
316
views
Are the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ always nontrivial in the bordism rings for $n>2$?
In my recent research, I need to know if the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ are always nontrivial in the unoriented and oriented bordism rings for $n>2$. (For the ...
- 587
12
votes
1
answer
406
views
Compact quaternionic Kahler manifolds of negative curvature: examples
There is a well known problem of LeBrun-Salamon:
are there any non-symmetric compact quaternionic-Kahler
manifolds of positive scalar (and Ricci) curvature?
It is hard and still unsolved:
Quaternionic-...
- 9,237
12
votes
0
answers
246
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: ...
- 2,516
12
votes
0
answers
241
views
Does cohomology ring determine a compact symmetric space?
Suppose that $M_1, M_2$ are compact connected symmetric spaces with isomorphic integer cohomology rings. Does it follow that $M_1$ is diffeomorphic to $M_2$?
The only result I am aware of is this ...
- 12.2k
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
11
votes
4
answers
2k
views
Hermitian symmetric spaces vs Hermitian homogeneous spaces
A Hermitian symmetric space is a connected complex manifold with a hermitian metric on which the group of holomorphic isometries acts transitively, and which satisfies the following extra condition: ...
- 2,713
11
votes
1
answer
536
views
Do there exist non-totally geodesic isometric minimal immersions $\mathbb{H}^2\rightarrow G/K.$
Suppose $G$ is a non-compact, semi-simple Lie group, of rank at least two, with maximal compact subgroup $K$ and $G/K$ the corresponding Riemannian symmetric space. Let $\mathbb{H}^2(-c^2)$ be the 2-...
- 3,020
11
votes
1
answer
433
views
Upper bounds for lattice points in orbits, and representations of binary quadratic forms
Write $\mathbb{Z}^{a\times b}$ for the $a\times b$ integer matrices. Let $n\geq 3$ and $Q\in\mathbb{Z}^{n\times n}$. Let $G=O(Q)$ be the orthogonal group of $Q$. For $X_0\in \mathbb{Z}^{2\times n}$, ...
10
votes
3
answers
489
views
Relationship between Multiplicative Ergodic Theorems
One version of Oseledets' Multiplicative Ergodic Theorem states that if $\sigma$ is an ergodic measure-preserving transformation of a space $(\Omega,\mathbb P)$ and if $A\colon\Omega\to GL(d,\mathbb R)...
- 23.2k
10
votes
1
answer
590
views
Involutive automorphism of simple Lie algebra
I am sorry if this question is too elementary to be posted here, but no experts answer this question when I post it on Math Stackexchange.
Let $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ be a Cartan ...
- 951
9
votes
2
answers
638
views
Curvature of the Cayley projective plane
The Cayley projective plane can be realized as the compact homogeneous space $F_4/\mathrm{Spin}(9)$. In this way one can compute the curvature of this symmetric space in terms of a suitable ...
- 403
9
votes
2
answers
386
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 ...
- 52.3k
9
votes
1
answer
643
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.
...
- 103
9
votes
0
answers
514
views
Volume of a geodesic ball in $\operatorname{SL}(n) / {\operatorname{SO}(n)}$?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}$Crossposted on MSE: https://math.stackexchange.com/questions/4261809/volume-of-a-geodesic-ball-in-sln-son
Question: What is the volume of a ...
- 637
9
votes
0
answers
326
views
Maximal geodesic spheres in the "octooctonic projective plane"
Boris Rosenfeld claimed that the 128-dimensional compact Riemannian symmetric space on which $\mathrm{E}_8$ acts as isometries could be seen as the "octooctonionic projective plane", $(\...
- 22.2k
9
votes
0
answers
268
views
Cohomology of $\operatorname{SO}(p,q;\mathbb{Z})$ with $p=3,q=19$
I would like to understand the topology of the moduli space of Einstein orbifold metrics on the $K3$-surface. It is known that this space is given by the bi-quotient $SO(3,19;\mathbb{Z})\setminus SO(3,...
- 423
8
votes
2
answers
459
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
8
votes
1
answer
247
views
Covolumes of unit groups of division algebras
Let $D$ be a central division (or maybe just simple) algebra over $\mathbb{Q}$. Let $\mathcal{O} \subset \mathcal{O}_m$ be an order inside a fixed maximal order and denote by $\mathcal{O}^1$ its group ...
- 767
8
votes
2
answers
452
views
Geodesic sphere in the octonion projective plane
I am considering Laplacian eigenvalues of a geodesic sphere in the octonion projective plane and I would like to know the metric on a geodesic sphere.
Does the metric on a geodesic sphere in the ...
- 141
8
votes
1
answer
387
views
What kind of locally symmetric space is a rational sphere
Using Dehn Surgery, we can construct compact hyperbolic $3$-manifolds with vanishing Betti numbers $b_1=b_2=0$, i.e., a rational homology $3$-sphere.
My question is the following.
Is there other ...
- 1,111
8
votes
2
answers
386
views
regular semisimple elements on spherical varieties
Let $(G,H_1)$ and $(G,H_2)$ be spherical pairs (i.e. $G$ is a reductive group, $H_i$ are its closed subgroups and the Borel subgroup $B$ of $G$ has a finite number of orbits on $G/H_i$).
What can ...
- 2,639
8
votes
1
answer
673
views
Classification of compact globally symmetric spaces
It is known that any connected compact Lie group $G$ is a finite quotient of the product of a compact simply connected semisimple Lie group $\tilde{G}$ and a torus $\mathbb{T}^n$ (see for example ...
- 193
8
votes
1
answer
650
views
Harish-Chandra isomorphism for compact symmetric spaces
I would be interested to have an explicit description of the algebra of invariant differential operators on functions on a compact symmetric space $G/K$. A reference would be especially useful.
Below ...
- 21.8k
7
votes
1
answer
226
views
Invariants for the isotropy representation of a Riemannian symmetric space
Statement: Let $M = G/K$ be a Riemannian symmetric space of compact type, and $V = T_o M$ be its isotropy representation (of $K$ acting on the tangent space of $M$). Then the Hilbert–Poincaré series $...
- 32.4k
7
votes
1
answer
1k
views
Help with definition of Liouville measure
$\require{AMScd}$For a Riemannian manifold $M$, I have read authors talking about a 'Liouville measure' on the unit tangent bundle $\operatorname{T}^1(M)$ and then proceed to claim/prove that it is ...
- 675
7
votes
2
answers
231
views
Non-semisimple symmetric subgroups of simply connected simple algebraic groups
Let $G$ be a simply connected simple algebraic group over the field of complex numbers $\mathbb C$. Let $H$ be a symmetric subgroup of $G$. This means that there exists an automorphism of order 2 $\...
- 14.1k
7
votes
1
answer
177
views
What are the manifolds whose Curvature tensor has a globally vanishing $k$th order covariant derivative
Let $(M,g)$ be a boundaryless Riemannian manifold whose curvature tensor have the property that there exists $k\geq 2$ such that $\nabla^k R\equiv0$. What is known about such Riemannian manfiolds ? Is ...
- 1,115
7
votes
3
answers
385
views
Equidistant hypersurfaces in symmetric space via exponentiation?
Here's some background and notation:
Let $G/K$ be a symmetric space of non-compact type. For concreteness, assume $G$ is in fact a classical simple real Lie group such as SL,SO, or Sp, and $K$ is a ...
- 1,329
7
votes
1
answer
208
views
An explicit description of neighborhoods of the rank 2 boundary in the Satake Compactification of $\mathbf{A}_2$
My Motivation: I'm having a hard time following the description of the topology in the Satake Compactification of locally symmetric spaces. The group theory is something I'm finding a bit tricky to ...
- 2,824
7
votes
0
answers
235
views
Algebraic version of Loos Symmetric Space
Ottmar Loos gave a definition of symmetric spaces in terms of the existence of a multiplication map. Namely, a manifold $M$ is symmetric if there exists a multiplication morphism $\mu:M\times M\to M$ ...
- 1,077
7
votes
0
answers
149
views
Eigenspaces and covering relations of twisted involutions
Let $\theta:G\to G$ be an involution of a complex connected reductive Lie group, preserving a maximal torus $T$ (which, for me, lies inside a $\theta$-invariant Borel $B$). Let $K = G^\theta$ be the ...
- 27.8k
7
votes
0
answers
507
views
Explicit formula for the Levi-Civita connection on a non-compact Riemannian symmetric space
Let $G/K$ be a non-compact Riemannian symmetric space, endowed with the Riemannian metric coming from the Killing form on the Lie algebra $\mathfrak{g}$ of the semi-simple Lie group $G$. Here $K$ is ...
- 1,942
7
votes
0
answers
138
views
Symmetric pairs of holomorphic type
Let $G$ be a real simple Lie group of Hermitian type; that is, $G/K$ carries a structure of a Hermitian symmetric space where $K$ is a maximal compact subgroup of $G$. Equivalently, the center $Z(\...
- 951
7
votes
0
answers
601
views
A few questions about $E_6$ and its symmetric spaces
Preface
The purpose of my question - on high level - is to understand exceptional symmetric spaces. My latest idea is to embed them into Lie group. There is quite nice embedding of 32-dimensional $E_{...
user21230
6
votes
2
answers
1k
views
Parallel forms and cohomology of symmetric spaces
Let $G/H$ be a compact symmetric space. Then I believe the following is true: if $\alpha \in \Omega^k(G/H)$ and $\nabla$ the Levi-Civita connection, then
$$
(\alpha \text{ is induced by an $\...
- 3,244
6
votes
1
answer
679
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 ...
- 821
6
votes
2
answers
430
views
Relationship between fans and root data
A (split) reductive linear algebraic group is equivalently described by combinatorial information called a root datum.
A toric variety is described by combinatorial information called a fan.
Both ...
- 4,081
6
votes
1
answer
395
views
Explicit fundamental domain for the action of $\operatorname{O}(n,1)(\mathbb{Z})$ on $\operatorname{O}(n,1)(\mathbb{R})$
Minkowski computed explicit fundamental domains for the action of $\operatorname{SL}_n(\mathbb{Z})$ on $\operatorname{SL}_n(\mathbb{R})/\operatorname{SO}_n(\mathbb{R})$ for each $n \leq 6$. In the ...