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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 ...
Vít Tuček's user avatar
  • 8,597
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 ...
asv's user avatar
  • 21.8k
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 ...
Tim Campion's user avatar
  • 63.9k
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 ...
DavidHume's user avatar
  • 743
2 votes
0 answers
225 views

Negative-curvature behaviour of higher-rank symmetric spaces

Let $X$ be a symmetric space of noncompact type with $rk\ X\geq 2$ and let $G$ be the identity component of the isometry group. Pick points $p\in X$ and $\xi\in\partial_{\infty}X$, with $\xi$ regular, ...
Elia Fioravanti's user avatar
3 votes
1 answer
266 views

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 $\partial X$,...
ThiKu's user avatar
  • 10.4k
2 votes
2 answers
464 views

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 ...
ThiKu's user avatar
  • 10.4k
21 votes
2 answers
1k views

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 ...
JME's user avatar
  • 3,022
6 votes
1 answer
589 views

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 ...
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