16
votes

Accepted

### Geodesic preserving diffeomorphisms of constant curvature spaces

For $\mathbb{R}^n$: the fundamental theorem of projective geometry (proof:
https://www3.nd.edu/~andyp/notes/FunThmProjGeom.pdf) says that the bijections of $\mathbb{R}^n$ taking lines to lines are ...

15
votes

Accepted

### Are the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ always nontrivial in the bordism rings for $n>2$?

There is a fibration $SU(n) \overset p\to SU(n)/SO(n) \overset j\to BSO(n)$, where the $j$ is the classifying map of $p$, viewed as (the projection of) a principal $SO(n)$-bundle. The Stiefel–Whitney ...

15
votes

Accepted

### Is there a contractible hyperbolic 3-orbifold of finite volume?

Yes. For example, let $M$ be the figure-eight knot complement. So $M$ is a hyperbolic manifold with volume a bit more than 2. The manifold $M$ has a two-fold symmetry $\tau$ that fixes, pointwise, a ...

12
votes

Accepted

### Is SO(2n+1)/U(n) a symmetric space?

Let me complement Claudio's answer. There is indeed a definition of symmetric space which works for any Riemannian manifold $M$: For any point $p\in M$ there is an involutive isometry $\iota_p$ of $M$ ...

12
votes

### Is SO(2n+1)/U(n) a symmetric space?

I think you are right to say that both homogeneous spaces are diffeomorphic as smooth manifolds. Therefore if one of them is a symmetric space, then one can transfer this structure to the other one.
...

12
votes

Accepted

### Definition of locally symmetric space of reductive groups

There is a very natural, intrinsic definition of a "symmetric space", as a manifold (Riemannian or Hermitian) with an extra symmetry of a certain prescribed type. It is then a theorem, not a ...

11
votes

Accepted

### Covolumes of unit groups of division algebras

First, you need to avoid definite quaternion algebras over $\mathbb{Q}$: in this case, the unit groups are finite, so the index cannot grow with $N$.
With that out of the way, your algebra $D$ ...

10
votes

### Almgren's regularity Theorem ; a simple example?

As I mentioned in my comment, a pair of orthogonal $2$-discs in $\mathbb{R}^4$ is known to be minimizing among all orientable $2$-currents with the same boundary by the technique of calibrations: One ...

9
votes

Accepted

### Explicit fundamental domain for the action of $\operatorname{O}(n,1)(\mathbb{Z})$ on $\operatorname{O}(n,1)(\mathbb{R})$

For information about these groups up through dimension 17, see:
Vinberg, È. B.
The groups of units of certain quadratic forms. (Russian)
Mat. Sb. (N.S.) 87(129) (1972), 18–36.
English translation [...

9
votes

Accepted

### Relationship between fans and root data

(1) A (connected) reductive group $G$ over an algebraically closed field $k$ is described by a combinatorial object called the based root datum ${\rm BRD}(G)$.
(2) A spherical homogeneous space $Y=...

9
votes

Accepted

### A different notion of a decomposable symmetric tensor (besides Veronese)

Your $\vee$ is essentially multiplication of polynomials. The variety of tensors $x_1 \vee \dotsb \vee x_m$ corresponds to polynomials that factor as products of linear factors. Points of the (...

8
votes

Accepted

### Compact quaternionic Kahler manifolds of negative curvature: examples

Any (Riemannian) symmetric space admits a cocompact lattice. This is due to A. Borel, Compact Clifford-Klein forms of symmetric spaces, Topology 2, 1963, pp.111-122. The quaternionic hyperbolic space ...

8
votes

Accepted

### Can all hermitian symmetric spaces be realised as coadjoint orbits?

This is true. One can use a few facts from Helgason's Differential Geometry, Lie Groups, and Symmetric Spaces to show that, indeed, $K = \mathrm{Stab}_G(Z)$.
Since $M=G/K$ is an irreducible Hermitian ...

7
votes

### Geodesic sphere in the octonion projective plane

Yes; every geodesic sphere $S_r$, $0<r<\pi/2$, in the octonion (or Cayley) projective plane $Ca P^2$ is isometric to a canonical deformation of the round metric on $S^{15}$ with respect to the ...

7
votes

Accepted

### The automorphism group of a symplectic symmetric space

The affine group of $(M,\nabla)$ is a Lie group $G$ by Kobayashi's theorem that shows that the automorphism group of any affine connection is a Lie group (see Kobayashi and Nomizu's Foundations of ...

7
votes

### Applications of Jordan algebras

Jordan algebras were originally introduced by Pascual Jordan in a hope to generalize the orthodox formulation of quantum mechanics, but this program was not successful as far as the generalization of ...

7
votes

### Jacobi fields on non-symmetric spaces

A particularly simple non-homogeneous example in which one can explicitly integrate the Jacobi equations is the complete metric on $\mathbb{R}^2$ given by
$$
g = (x^2{+}y^2{+}2)\bigl(\mathrm{d}x^2+\...

7
votes

### Spherical roots, restricted roots, and the dual group of a symmetric variety

There is a small survey published in an Oberwolfach report by Bart Van Steirteghem comparing the different normalizations of spherical roots in more detail. The standard normalization is obtained by ...

6
votes

Accepted

### Relationship between Multiplicative Ergodic Theorems

The equivalence of "sublinear shadowing" property and the Oseledec theorem (or, rather, of what is called "Lyapunov regularity") is due to Kaimanovich MR0947327 (89m:22006), and is explained there in ...

6
votes

### When can the metric be reconstructed (up to scaling) from knowing the conjugate points?

I guess the most famous cases are: (1) Riemannian tori without conjugate points are flat (Hopf, Burago and Ivanov) and (2) Auf-Wiedersehen metrics on the sphere (Green, Berger) are round (i.e. your ...

6
votes

Accepted

### Holonomy groups of compact Riemannian symmetric spaces

At the request of the OP I put my comment as an answer: in general, the holonomy group and the isotropy group have the same identity component (this is a theorem of E. Cartan). So if you assume that $...

6
votes

### Bounded non-symmetric domains covering a compact manifold

A silly generalization would be a direct product of Koadaira's surface and, say, a Riemann surface. A better construction is below.
I will use
Griffiths, Phillip A., Complex-analytic properties of ...

6
votes

Accepted

### Invariants for the isotropy representation of a Riemannian symmetric space

One reference is in Helgason's 1984 book Groups and Geometric Analysis. The result you want appears there as Corollary 5.12.
The notation he uses is $X=G/K$ is a symmetric space where $G$ is ...

6
votes

### Is there a contractible hyperbolic 3-orbifold of finite volume?

As pointed out by Moishe Kohan in the comments below, the following doesn't answer the question as asked, because my group $\Gamma$ is not contained in $SO(3,1)$. Anyway, here is an easy description ...

6
votes

Accepted

### Buildings as generalizations of symmetric spaces

(Wanted to post as a comment but didn't have enough reputation)
I have a partial answer only for your first question.
Let $F$ be a non-Archimedean field. For p-adic(or $\pi-$adic) symmetric space $\...

6
votes

### References for $K$-orbits in $G/B$

A good reference for this is "On Rationality Properties of Involutions of Reductive Groups" by Helminck and Wang (Advances in Math. 1993). The decomposition of $K\backslash G/B$ is studied ...

5
votes

Accepted

### Irreducible Symmetric Pairs

It is true that $\mathfrak p$ is always an irreducible $\mathfrak k$-module. Be aware though that the complexification $\mathfrak p_{\mathbb C}$ might be reducible as an $\mathfrak k_{\mathbb C}$-...

5
votes

Accepted

### The uniqueness of a $K$-fixed vector in a spinor representation

There is no contradiction. To use your physical language, the fully filled state is not $K$-fixed as a vector. Rather, the group ${\rm U}(n)$ acts on it by the determinantal character. The subtle ...

5
votes

Accepted

### Help with definition of Liouville measure

The construction doesn’t really simplify on symmetric spaces. On $TM\cong T^*M$ (using the metric) consider the canonical 1-form $\alpha=“\langle p,dq\rangle”$ and symplectic form $d\alpha$ and ...

5
votes

Accepted

### Cut locus for simply connected manifolds

The equality $D_p = C_p$ holds for compact symmetric spaces (CROSSes) of rank $1$, but not in general for higher rank symmetric spaces.
A rank $1$ CROSS is isometric to a round sphere or projective ...

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