All Questions
9 questions
3
votes
0
answers
142
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 ...
- 954
5
votes
0
answers
276
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
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
3
votes
1
answer
369
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
1
vote
0
answers
101
views
Actions of finite groups on compact symmetric spaces
I am interested in series of finite subgroups of the classical compact simple Lie groups which have big orbits on compact symmetric spaces and where the double coset space has some nice explicit ...
- 8,597
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-...
- 5,405
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
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
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