Questions tagged [isometry-groups]
Questions about the group of isometries of a metric space, in particular, a Riemannian manifold.
20
questions
4
votes
0
answers
60
views
Isometry group of the Fubini-Study metric on complex projective spaces
Let $(\mathbb CP^n,g_{FS})$ be the complex projective space equipped with the standard Fubini-Study metric.
What is the Riemannian isometry group of $(\mathbb CP^n,g_{FS})$? It seems to me that its ...
- 1,401
8
votes
0
answers
164
views
Involution of 3-sphere
Suppose that a closed geodesic $\gamma$ is the fixed-point set of an isometric involution on $(\mathbb{S}^3,g)$. Assume that sectional curvature of $g$ is at least $1$.
Is it true that $$\mathrm{...
- 44.4k
0
votes
0
answers
46
views
On sub-maximally symmetric Riemannian spaces
Is there a 4-dimensional Riemannian manifold with 8-dimensional isometry group?
Context: Guido Fubini (Annali di Mat., ser. 3, 8 (1903) 54) shows that the dimension $n$ of the isometry group of a $d$-...
- 427
2
votes
0
answers
79
views
On the convergence of isometries
Proposition: Let $(M,g)$ be a closed, compact Riemannian manifold. Let $f_n:M\to M$ be a sequence of isometries satisfying
$$
\|f_n - \iota\|_{L^2} \le \frac{1}{n},
$$
where $\iota:M\to M$ is the ...
- 705
2
votes
0
answers
74
views
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 $\...
- 479
5
votes
1
answer
285
views
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 ...
- 44.4k
10
votes
1
answer
556
views
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\...
- 12.8k
3
votes
0
answers
486
views
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 ...
- 31
14
votes
2
answers
908
views
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 ...
- 455
2
votes
0
answers
73
views
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 ...
- 877
1
vote
1
answer
164
views
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?
- 1,401
4
votes
0
answers
95
views
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 ...
- 213
5
votes
1
answer
172
views
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 $f(\gamma)$.
...
- 147
4
votes
1
answer
188
views
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,\...
- 213
5
votes
0
answers
170
views
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 ...
