Questions tagged [isometries]
The isometries tag has no usage guidance.
26 questions with no upvoted or accepted answers
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Can a closed disc in the plane be partitioned into three disjoint sets which are pair-wise isometric?
Any progress on the following: Can a closed disc in the plane be partitioned into three disjoint sets which are pair-wise isometric, i.e. each set is an image of the others under an isometry?
- 700
7
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Homometric $\Rightarrow$ isometric?
Suppose you know that there is a mapping between
two Riemmanian manifolds $M_1$ and $M_2$ such that,
for each $x_1 \in M_1$, the (codimension-1) measure of the set of points
at distance $d$ from $x_1$ ...
- 151k
6
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What is the minimum $n$ for which $\Bbb H^3$ can be isometrically embedded in $\Bbb R^n$ as a bounded set?
Consider the hyperbolic $3$-space $\Bbb H^3$ (i.e., the unique, simply-connected, $3$-dimensional complete Riemannian manifold with a constant negative sectional curvature equal to $-1$). The Nash ...
- 1,097
6
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691
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Isometries of Compact Semisimple Lie Groups
In this delightful question, the poster mentioned that the isometry group of a compact Lie group $G$, equipped with the metric from the Killing form, is $G\times G/Z(G)$, where $Z(G)$ is the center of ...
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4
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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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4
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0
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77
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Proximal isometries in CAT($-1$) metric space
Let $X$ be a rank $1$ symmetric space of non-compact type and $G$ its isometry group. $G$ is a semisimple linear algebraic Lie group of non-compact type with trivial center. Let $\rho$ be a ...
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3
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0
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Isometric embeddings of $c_0$ into metric spaces
Are there any nice and useful criteria or theorems which assert when a given metric space $M$ contains an isometric (not necessarily linear) copy of the Banach space $c_0$ or its unit ball $B_{c_0}$? (...
- 1,151
3
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0
answers
104
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Every partial isometry extends
I am interested in metric spaces $X$ where every isometry between two subsets of the space extends to a full isometry $X \to X$. Is there a name for this kind of space? Is there some paper which ...
- 31
2
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0
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320
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What are examples of "perfect tensors"?
A "perfect tensor" is defined on the nLab very abstractly as "its tensor/hom-adjuncts $V^{\otimes k} \to V^{\otimes n - k}$ for $k \le n/2$ are isometries". The only example I'm ...
- 451
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Does a lifted functor on $\mathbf{1Met}$ preserve isometries?
Let $\mathbf{1Met}$ denote the category of metric spaces with distance bounded by $1$ and nonexpansive maps ($1$-Lipschitz functions). I call isometry a distance-preserving map (some people require it ...
- 93
2
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150
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Spacetime symmetries
We know some nice space-time have a lot of symmetries. It is said that
Minkowski spacetime has
$$ISO(d-1,1)/SO(d-1,1),$$
de Sitter spacetime has
$$SO(d,1)/SO(d-1,1)$$ and
anti-de Sitter spacetime ...
- 2,147
2
votes
0
answers
207
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Minkowski isometries
Consider theorem 1.7 from chapter III of 'Elementary differential geometry' by O'Neill. It says that:
Theorem 1.7: If $\phi$ is an isometry of $E^3 $, then there exists a unique translation $T$ and a ...
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2
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0
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187
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The isometry groups of flag manifolds
For any sequence of integers $0<n_1<...<n_k$, there is a flag manifold of type $(n_1, ..., n_k)$, which is the collection of ordered sets of vector subspaces of $R^{(n_k)}$ $(V_1, ..., V_k)$ ...
- 10.5k
2
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0
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42
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On the minimum distance along an orbit
Let $\Gamma$ be a nontrivial group of isometries of $\mathbb{S}^n$, $n \geq 2$, acting properly discontinuously. For $p \in \mathbb{S}^n$, define
$$r(p) = \min_{g \in \Gamma \setminus\{e\} } d(p, g(p)...
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1
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0
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Instantaneous rotation field in relation to a developable surface
I have a ruled surface, let it be given by $\Sigma: U \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ parametrized by $(u,v)$ with the rulings along the $u$-lines. Now, let $X: U \subset \mathbb{R}^2 \...
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1
vote
0
answers
143
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Ideal Ford domain (for finite index subgroup)
Let $G$ be a lattice Fuchsian group with parabolic elements, seen as a discrete subgroup of matrices
$
g=
\begin{pmatrix}
\alpha & \overline{\beta} \\
\beta & \overline{\alpha}
\end{pmatrix}
$...
- 33
1
vote
0
answers
187
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Does there exist an isometry between a regular polygon and a circle?
In order to define the question in a meaningful fashion, I am referring to a smooth manifold $\mathcal{M}$ within an $\epsilon$-neighborhood of a regular polygon $\mathcal{P}$ satisfying $$\max\{\|x-p\...
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1
vote
0
answers
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Mapping to distorted constant Gauss curvature surfaces of revolution
There are three questions here. We imagine a flexible membrane that is scrolled out so as to straighten it.
1) How can we find a surface isometrically mapped from a surface of constant negative Gauss ...
- 917
1
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0
answers
561
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Bending Beltrami Pseudosphere
The Beltrami Pseudosphere
$$[x = a \sin p \cos t , y= - a ( \cos p + \log \tan p/2 ) , z= b+ a \sin p \sin t \; ], (.1 <p<\pi/2), (0< t< 2 \pi), \; (b>a) $$
is bent to a non-...
- 917
1
vote
0
answers
82
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Finding the infimum using a piecewise isometry
Given a finite set of unit circles in the plane such that the area of their union $U$ is $S$, what is the largest possible bound $kS$ for some constant $k$ such that there exists a subset of mutually ...
- 201
1
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0
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393
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Surjectively isometric normed spaces: Hamel vs (extended) Schauder dimension
Bonjour/bonsoir à toutes et à tous.
This may really be a very basic question, but... Let $\mathbf{X} \equiv (X, \|\cdot\|_X)$ and $\mathbf{Y} \equiv (Y, \|\cdot\|_Y)$ be surjectively isometric (1) ...
- 10.5k
1
vote
1
answer
345
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What is general expression for the moment map of a Kaehler Hamiltonian G-manifold
A Kaehler Hamiltonian G-manifold is a Kaehler manifold with a Hamiltonian G-action, i.e., a G-action generated by a moment map. In particular, the Killing vector fields which generate the G-action are ...
- 1,155
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Proving that Euclidean Jordan algebra automorphisms are orthogonal symmetric cone automorphisms
Recall the setting of,
Jacques Faraut and Adam Korányi. Analysis on Symmetric Cones. Clarendon Press, Oxford, 1994. ISBN 9780198534778.
Namely, suppose that,
$\left(V,\circ,\left<\cdot,\cdot\...
- 111
0
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0
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109
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Commutator subgroup of rotational symmetries of the hypercube
I would like to know which is the commutator subgroup of the group of rotational isometries of the $n$-dimensional cube. The group i am talking about is the subgroup of $\{ \pm 1 \} \wr S_n$ ...
0
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0
answers
148
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Symmetries higher dimensional cube fixing subcubes
Which is the group of rotational isometries of an $n$ dimensional hypercube fixing an $m$ dimensional element (an $m$ dimensional subcube)? I know for example that it is $A_n$ for $m=0$ (symmetries ...
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1k
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Surface locally isometric to a sphere.
If for any two points p,q in a regular, compact surface $S\subseteq R^3$, there exists an isometry $f:S\rightarrow S$ s.t. f(p)=q. How to prove that $S$ is locally isometric to the sphere?
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