# Questions tagged [isometry-groups]

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

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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 ...
1 vote
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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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### 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 ...
612 views

### 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?
1 vote
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 ...