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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 ...
• 25.6k
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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 ...
• 2,984
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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 ...
• 26.4k
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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 $M$ ...
• 15.3k

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. ...
• 4,674
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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 ...
• 36.2k
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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 $D$ ...
• 2,949

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 ...
• 106k
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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 [...
• 591
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• 37.3k

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 ...
• 31k
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 ...
• 106k
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 ...