Questions tagged [isometries]

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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?
James Currie's user avatar
7 votes
0 answers
645 views

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 $x_1$ ...
Joseph O'Rourke's user avatar
6 votes
0 answers
146 views

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 ...
Random's user avatar
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5 votes
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619 views

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 ...
Robin Goodfellow's user avatar
4 votes
0 answers
91 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 ...
Dinisaur's user avatar
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4 votes
0 answers
77 views

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 ...
user470881's user avatar
3 votes
0 answers
109 views

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 $B_{c_0}$? (...
Damian Sobota's user avatar
3 votes
0 answers
97 views

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 ...
James's user avatar
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2 votes
0 answers
252 views

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 ...
unknown's user avatar
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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 ...
ralphS16's user avatar
2 votes
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139 views

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 ...
annie marie cœur's user avatar
2 votes
0 answers
196 views

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 ...
Soennecken's user avatar
2 votes
0 answers
180 views

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 $R^{(n_k)}$ $(V_1, ..., V_k)$ ...
wonderich's user avatar
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2 votes
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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)...
Eduardo Longa's user avatar
1 vote
0 answers
77 views

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 \...
RWien's user avatar
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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} $...
user178149's user avatar
1 vote
0 answers
171 views

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\...
Talmsmen's user avatar
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1 vote
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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 ...
Narasimham's user avatar
1 vote
0 answers
555 views

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-...
Narasimham's user avatar
1 vote
0 answers
82 views

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 ...
user19405892's user avatar
1 vote
0 answers
392 views

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) ...
Salvo Tringali's user avatar
1 vote
1 answer
331 views

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 ...
Mtheorist's user avatar
  • 1,135
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0 answers
22 views

Characterizing isometry invariant functions

I am trying to understand functions invariant under isometries better. In particular functions with two inputs. I.e. $$ C(x,y) = C(\phi x, \phi y) $$ for any isometry $\phi$. The motivation are ...
Felix B.'s user avatar
  • 347
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0 answers
107 views

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 $\{ \pm 1 \} \wr S_n$ ...
quelramodellago's user avatar
0 votes
0 answers
147 views

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 ...
Giovanni_M's user avatar
0 votes
0 answers
1k views

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?
diego0's user avatar
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