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
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Cusps as warped products
It is well-known that the ends of a finite-volume hyperbolic manifold are warped products $$(0,\infty)\times_f T$$
for some euclidean manifold $T$ and $f(t)=e^{-t}$.
Question: Is there a similar ...
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Are harmonic maps quasiconformal at the boundary of hyperbolic spaces?
Let $M$ be a hyperbolic $3$-manifold and $X$ a negatively curved, simply connected space with geometric boundary $\partial X$.
If $\rho:\pi_1M\rightarrow Isom(X)$ does not fix a point in $\partial X$,...
- 10.4k
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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 ...
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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 ...
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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 ...
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Is SL_n/S(GL_k x GL_n-k) symmetric?
Background: a symmetric variety is a homogeneous space $G/H$ associated to an involution $\theta$ of a semisimple algebraic group $G$ and $\{g | \theta(g) = g\} = G^\theta \subset H \subset N_G(G^\...
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Seeking a generalization of group embedding of symmetric varieties
I am looking for generalizations of the following construction.
Let $G$ be a connected, reductive group and let $\theta : G \rightarrow G$ be an involution. Let $H = G^{\theta}$ be the subgroup of $\...
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Ricci-flat Kähler metrics on symmetric varieties
Hallo,
I have a question on a paper of Azad and Kobayashi "Ricci-flat Kähler metrics on symmetric varieties". Here is the link: http://www.academia.edu/2579043/Ricci-...
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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 $\...
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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 ...
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Combination theorems for discrete subgroups of isometry groups
Maskit's combination theorem says: if $M=M_1\cup_\Sigma M_2$ is a union of hyperbolic 3-manifolds $M_1=\Gamma_1\backslash H^3, M_2=\Gamma_2\backslash H^3$ along a surface $\Sigma$, and if the limit ...
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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 ...
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Linear symmetric spaces are spaces with ''orthogonal complements''?
The collection of marked unimodular lattices in the euclidean $n$-space corresponds to the symmetric space $S_n:=SO_n(\mathbb{R}) \backslash SL_n(\mathbb{R})$.
I have only recently been made aware ...
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Chains in $K\backslash G/B$ lying over a closed $K$-orbit
Let $G$ be a complex connected reductive Lie group, $\theta$ an
involution, and $K = G^\theta$ the fixed-point subgroup.
Then $K$ has finitely many orbits on $G/B$, one of which is open
and (quite ...
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Covering relations in $K\backslash G/B$
Let $G$ be a simply connected complex Lie group, $\theta$ an involution,
and $K = G^\theta$ the fixed point subgroup. Pick a $\theta$-invariant
Borel subgroup $B$. Then there is a natural
map $K\...
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volume form in a symmetric space of real rank one
I want to compare the two canonical volume forms on a noncompact symmetric space fo real rank one.
The first one is the volume form induced by the Riemannian structure given by the Killing form ...
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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$$ the ...
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The relative homotopy sets $\pi_1(U_n,U_n/O_n)$ and $\pi_1(U_{2n},U_{2n}/USp_{2n})$
During my research I have recently stumbled upon the problem of finding the relative homotopy sets $\pi_1(U_n,U_n/O_n)$ and $\pi_1(U_{2n},U_{2n}/USp_{2n})$ for $n$ large enough to be in the stable ...
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Volume of fundamental domain and Haar measure
In my research, I do need to know the Haar measure. I have spent some time on this subject, understanding theoretical part of the Haar measure, i.e existence and uniqueness, Haar measure on quotient. ...
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Names of noncompact riemannian symmetric spaces?
Irreducible riemannian symmetric spaces come in pairs: one compact and one not compact, usally called the noncompact dual.
The compact symmetric spaces include spheres, complex and quaternionic ...
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different Shimura data with common underlying group?
A pure Shimura datum is of the form $(G,X)$ with $G$ a connected reductive $\mathbb{Q}$-group, and $X$ a homogeneous space under $G(\mathbb{R})$, subject to Deligne's conditions in terms of Hodge ...
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Is the Baily--Borel compactification functorial?
The following question seems pretty natural, but searching online
and looking in some obvious places didn't turn up much, so maybe
I can ask it here. (Disclaimer: I'm a newcomer to this topic, so
...
user5117
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answers
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A question about the affine Grassmanian
For $SL(2, \mathbf{C} ((t)))$ the affine Grassmanian is defined as:
$$SL(2, \mathbf{C}((t))) / SL(2, \mathbf{C} [[t]])$$
Now that is fine but $SL(2, \mathbf{C} ((t)))$ has smaller parabolic subgroups. ...
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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 ...
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links and interactions between different approaches to (super-)rigidity
By super-rigidity I mean some theorems concerning the arithmetic subgroups in semi-simple Lie groups. According to Margulis "Discrete subgroups of semi-simple Lie groups" (the book published by ...
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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 ...
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question about equivariant embeddings of riemannian symmetric domains
Here by riemannian symmetric domain is understood an riemannian symmetric space with only factors of non-compact types. Such domains are realized as quotients of the form $D=G/K$, where $G$ is a ...
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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 ...
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Do symmetric spaces admit isometric embeddings as intersections of quadrics?
While preparing a seminar I gave today, the following question arose. I asked the seminar participants, but nobody knew the answer. Hence I'm asking it here in MO.
Background
Recall that a ...
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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: ...
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Generalizing cosine rule to symmetric spaces
The sine and cosine rules for triangles in Euclidean, spherical and hyperbolic spaces can be understood as invariants for triples of lines. These invariants are given in terms of the distance (both ...
user2529
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"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. ...
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