# Questions tagged [isometry-groups]

Questions about the group of isometries of a metric space, in particular, a Riemannian manifold.

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### Decompositions of groups and the existence of apartments

Let $X$ be an affine building and $G$ a group with isometric action on $X$. For any non-empty subset $\Omega$ of $X$, we denote by $P_{\Omega}$ the fixer of $\Omega$. Similarly, for any sector $\...

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### All-set-homogeneous spaces

This is a follow-up to the question of Joseph O'Rourke Which metric spaces have this superposition property?
A metric space $X$ will be called all-set-homogeneous if for any subset $A\subset X$ any ...

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### Does a compact contractible metric space have a point that is fixed by all isometries?

Let $(X,d)$ be a compact and contractible metric space. Let $\operatorname{Isom}(X)=\{\phi\colon X\to X\}$ be its group of isometries.
Question: Is there a point $x\in X$ fixed by all $\phi\in\...

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### Isometries of the complex projective space for the Fubini Study metric

$\DeclareMathOperator\SU{SU}$I am trying to understand a geometric proof in our mathematical quantum mechanics lecture regarding Wigner's theorem in finite dimensions. We have already shown that it ...

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### Quasi-isometry groups of metric spaces

Given a metric space $(X, d)$, we can consider the set of all quasi-isometries $f: X \to X$, and quotient out by the equivalence relation identifying $f$ and $g$ if $\sup_{x \in X}d(f(x), g(x))$ is ...

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### Reference request: Discrete subgroup of $\mathrm{PO}(n,1)$ preserving proper subspace has infinite covolume

I'm looking for a reference for the following claim:
$\newcommand{\PP}{\mathrm{PO}(n,1)}$
Let $\PP$ denote the group of isometries of $V = \mathbb{R}^{n,1}$ preserving the upper sheet of the ...

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### Classification of the group action

Let $G$ be a closed subgroup of $O(n)$ such that $\mathbb R^n/G$ is isometric to $\mathbb R^{n-2} \times \mathbb R_+$. Can we have a classification of $G$ up to conjugation?

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### Isometries of fiber bundles

Let $F\to S\overset{\pi}{\to} B$ a Riemannian submersion with totally geodesic fibers.
Question: How much information about the isometries of $S$ we have if we know the isometries of $F$ and $B$? For ...

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### Angle between geodesics at different fixed points of a Riemannian isometry

Suppose I have an isometry $f$ of a Riemannian manifold $(M,g)$. Suppose further that $p$ and $q$ are fixed points of $f$. If $\gamma$ is a geodesic segment from $p$ to $q$, then so is $f(\gamma)$.
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### An explicit description of $\operatorname{Isom}(\widetilde{\operatorname{Sl}_2})$

$\DeclareMathOperator\Sl{Sl}\DeclareMathOperator\PSl{PSl}\DeclareMathOperator\Isom{Isom}$Let $\widetilde{\Sl_2}$ be the Thurson geometry that can either be described as the universal cover of $\PSl(2,\...

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### Correspondence between Riemannian metrics and Euclidean embeddings

Given a sufficiently smooth manifold M,
a Riemannian metric on M induces an isometric embedding into Euclidean space by Nash's theorem, (non-canonically, non-uniquely)
an embedding of M into ...

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### Let $G'\triangleleft G<\operatorname{Iso}(M)$ be a normal subgroup. A $G'$-stratum is the union of $G$-strata of lesser dimension

Let $G$ be a group of isometries acting effectively by isometries on a connected Riemannian manifold. And let $G'\triangleleft G$ be a normal subgroup. I am trying to prove that $\dim \operatorname{St}...

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### The isometry group of a product of two Riemannian manifolds

Under what conditions is the isometry group of a product of two Riemannian manifolds the product of the isometry groups of each one of the components?
One counterexample is a product of two isometric ...

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### Possible isometry groups of open manifolds

Consider a non-compact manifold $M$.
Does there always exist a Riemannian metric on $M$ such that the isometry group is non-compact?

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### Can Riemann's explicit formula be generalized to semi-primes?

Following Isometry group of an integer I wonder if one can define a "mock zeta function" $\zeta_{V}$ (where $V:=(\mathbb{Z}/2\mathbb{Z})^{2}$ stands for "Vierergruppe", the German word for the Klein ...

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### Closed Semi-Riemannian manifolds with non-compact isometry group

Does anyone know a good reference for general results about closed Semi-Riemannian manifolds which have a non-compact isometry group?
Edit: My goal is to understand a bit better what the intuition ...