All Questions
Tagged with symmetric-spaces riemannian-geometry
41 questions
3
votes
1
answer
100
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
0
answers
165
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) ...
- 1,097
1
vote
0
answers
35
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 ...
- 1,879
1
vote
0
answers
95
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
3
votes
1
answer
273
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 $...
- 759
5
votes
1
answer
169
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 ...
- 4,081
6
votes
0
answers
147
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 $. ...
- 4,081
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
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
2
votes
1
answer
294
views
Complex quadric as a symmetric space
It is known that a smooth complex quadric is a symmetric space. For example, it is
$$\operatorname{Spin}(n+2)/G$$
where $G$ is the maximal parabolic subgroup.
I want a reference for more details and ...
- 21
2
votes
0
answers
74
views
Image of tori in locally symmetric spaces and homology
Suppose we have a reductive group $G$ over $\mathbb{Q}$, a compact subgroup of the adelic points $K_f\subset G(\mathbb{A}_{\mathbb{Q}})$, and the associated locally symmetric space
$$Y_K := G(\mathbb{...
- 2,044
5
votes
0
answers
275
views
Fundamental group of compact globally symmetric spaces
The fundamental group of a globally symmetric space $M$ of compact type is known (see Loos [1], Borel [2]). The result can be formulated as follows: it is isomorphic to the quotient
$$(*) \quad \pi_1(...
- 1,123
4
votes
1
answer
133
views
Rigidity of the compact irreducible symmetric space
Let $(M^n,g)$ be an irreducible symmetric space of compact type. In particular, $(M^n,g)$ is an Einstein manifold with a positive Einstein constant.
Is there any classification for $(M^n,g)$ if $(M^n,...
- 891
5
votes
1
answer
184
views
Proof of equivalence between Lie triple systems and totally geodesic submanifolds
In a Riemannian symmetric space $Q$, it is well known that the existence of a totally geodesic submanifold at a point $p \in Q$ is equivalent to the existence of a Lie triple system at $p$, i.e., a ...
- 965
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
5
votes
2
answers
379
views
Correct curvature tensor of symmetric space of positive definite matrices with trace metric?
Let $Pos(n)$ be the set of $n \times n$ real positive definite matrices with trace (aka affine-invariant) metric
$$\langle u, v \rangle_p = tr(p^{-1} u p^{-1} v)$$
for all $p \in Pos(n)$ and $u, v \in ...
- 637
3
votes
1
answer
367
views
Closed manifolds of nonnegative curvature operator are symmetric spaces
In an online webinar, I heard (not directly) the statement that (closed) manifolds of nonnegative curvature operator $\mathcal{R}\geq 0$ are symmetric spaces. Is this a valid theorem? Any reference ...
- 4,195
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
4
votes
1
answer
262
views
Upper bounds on the sectional curvature of the real Grassmannian
Consider the real Grassmannian as the symmetric space $\operatorname{Gr}(n,k) \cong \operatorname{O}(n)/(\operatorname{O}(k) \times \operatorname{O}(n-k))$ for $n \geq 3$, $k \geq 2$, where the metric ...
- 177
1
vote
1
answer
122
views
how to construct a finite energy map
In the construction of harmonic maps by Eells and Sampson, one needs to start with a map with finite energy and use the heat equation to deform it into a harmonic map. The construction of such a ...
- 68
2
votes
0
answers
67
views
Closed-form expression for Riemannian exponential maps on symmetric spaces
Besides the Poincaré model for the hyperbolic disc, $S^n$, and Euclidean space, what are known instances of a symmetric space $M$ (finite-dimensional) for which the exponential map is known in closed-...
- 5,407
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
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
2
votes
1
answer
328
views
Totally geodesic submanifolds of bi-invariant Lie groups
Let $G$ be a Lie group equipped with a bi-invariant metric. I have some questions concerning the totally geodesic and the flat (all sectional curvatures zero) submanifolds of $G$.
I known that every ...
- 965
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
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
4
votes
0
answers
114
views
Representation theoretic characterisation of symmetric spaces
Let $G$ be a simple compact Lie group and $H$ a closed subgroup.
Let $\mathfrak{h}\subset \mathfrak{g}$ denote the corresponding Lie algebras. Let $\mathfrak{m}$ be an orthogonal complement to $\...
- 81
2
votes
0
answers
41
views
Cylindrical coordinates in quotient of symmetric space
I am interested in the following situation. Suppose $G/K$ is a symmetric space of non-compact type and $\alpha$ is the axis of a hyperbolic isometry. I am interested in computing the Hessian of the ...
- 445
2
votes
1
answer
93
views
Homogeneity of a projective vector bundle
Let $M=G/K$ be a $G$-homogeneous manifold and suppose that $E\to G/K$ is a homogeneous (complex) vector bundle, i.e. it is defined by a representation $\phi : K \to \text{Aut(V)}$ for some complex ...
- 21
5
votes
1
answer
206
views
When can the metric be reconstructed (up to scaling) from knowing the conjugate points?
Let $M$ be a smooth manifold of dimension $\geq 2$. Let $g$ be a complete Riemannian metric on $M$. Let $C \subseteq M \times M$ be the set of pairs of $g$-conjugate points.
The set $C$ doesn't ...
- 63.9k
6
votes
1
answer
465
views
Holonomy groups of compact Riemannian symmetric spaces
Let $M$ be a compact Riemannian symmetric space. By the classification of Cartan, it belongs to the table of homogeneous spaces given in the Wikipedia page:
https://en.wikipedia.org/wiki/...
- 643
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
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
5
votes
0
answers
200
views
Volume growth and visibility in hyperbolic spaces
I am interested in the following two questions for complex, quaternionic and octonionic hyperbolic spaces, equipped with their usual metrics and measures:
For brevity, I will denote the volume of the ...
- 743
2
votes
0
answers
82
views
symmetric points on symmetric spaces
Let $M$ by an $m$-dimensional symmetric space (or a general Riemannian manifold). The finite distinct points $p_1,p_2,\cdots,p_n\in M$ are said symmetric, if for any permutation $\sigma$ on $1,2,\...
- 543
4
votes
1
answer
253
views
Non-flat totally geodesic surfaces
I'd like to know whether a Riemannian symmetric space of compact type admits a non-flat totally geodesic surface. I've found an article by Mashimo on the classification of these surfaces for certain ...
- 1,378
4
votes
1
answer
351
views
Volume of arithmetic quotients of symmetric spaces
Now let $\textbf{G}$ be some connected semisimple linear algebraic group over a number field $F$. Let $G_{\infty}$ be $\textbf{G}(\mathbb{R}\otimes_{\mathbb{Q}} F)$. Let $K_{\infty}$ be a maximal ...
- 685
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
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
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
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