Questions tagged [isometries]
The isometries tag has no usage guidance.
100
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If $X,X'$ have the same $\varepsilon$-packing numbers and $f:X \to X'$ surjective $1$-Lipschitz, then $f$ is an isometry
Let $(X, d)$ be a compact metric space.
We say that $\{x_1, \cdots, x_n\} \subseteq X$ is an $\varepsilon$-covering of $X$ if for any $x \in X$, there exists $i \in \{1, \ldots, n\}$ such that $d(x, ...
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Are two metric spaces isometric if they have the same $\varepsilon$-covering and $\varepsilon$-packing numbers for all $\varepsilon>0$?
Let $(X, d)$ be a compact metric space.
We say that $\{x_1, \cdots, x_n\} \subseteq X$ is an $\varepsilon$-covering of $X$ if for any $x \in X$, there exists $i \in \{1, \ldots, n\}$ such that $d(x, ...
- 713
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Are two metric spaces isometric if they have the same $\varepsilon$-covering numbers for all $\varepsilon>0$?
Let $(E, d)$ be a metric space. For $\varepsilon>0$, we define two notions of $\varepsilon$-covering number as follows, i.e.,
$N_\varepsilon^o (E)$ is the smallest number of open balls whose radii ...
- 713
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1
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186
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Is every 1-Lipschitz homeomorphism $f:X\to X$ from a compact metric space to itself an isometry?
I found a statement involving a homeomorphism $f:X\to X$ of a compact metric space $X$, with Lipshitz coefficient 1, i.e., a non-expansive map, and cannot think of an example where $f$ is not an ...
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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 ...
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139
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Isometries of Hilbert space
It is easy to see that for any $x$ and $y$ on the unit sphere of a Hilbert space $H$ there exists a surjective isometry $U$ such that $Ux=y$. Does something more general also hold? That is, given two ...
- 1,157
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1
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181
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Are the ideals in two $C^*$-algebras the same?
Let $V_{1}, V_{2}$ be the commuting isometries. By Wold decomposition theorem, we know that $V_{i}$ admits decomposition $$V_i \cong V^s_{i}\oplus V^{u}_{i},$$ where $V^{s}_{i}$ is the shift and $V^{u}...
- 139
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Does the isometry group determine the Riemannian metric?
Suppose $G \subset \text{Iso}(M)$ is a Lie group acting smoothly on a (pseudo-)Riemannian manifold $(M, g)$. Then $G$ induces a Lie algebra of Killing vectors on $M$. In this paper by Goenner and ...
- 203
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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 ...
- 41k
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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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A space isometric to $\ell_\infty^2$
Consider a norm on $\mathbb C^2$ as $\|(z_1,z_2)\|:=\max\{|z_1|,|z_2|,\frac{1}{\sqrt{2}}|z_1+iz_2|\}.$
Question. Is $(\mathbb C^2,\|.\|)$ linearly isometric to $(\mathbb C^2,\|.\|_{\infty})$ where $\|(...
- 1,645
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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 $B_{c_0}$? (...
- 1,141
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Are polyhedra with equilateral triangular faces rigid?
Convex polyhedra are rigid by Cauchy’s theorem. Steffen’s polyhedron is an example of a non-convex polyhedron that is flexible (i.e., non-rigid). However, it appears to have edges of different lengths....
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Isometry between Minkowski space and Tangent space in an article by Stefan Waldmann [closed]
In the notes Geometric Wave Equations by Stefan Waldmann at page 70 they have
Having a fixed Lorentz metric $g$ on a spacetime manifold $M$ we can
now transfer the notions of special relativity, ...
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2
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Fixed points on spherical buildings
A crucial aspect of the Bruhat–Tits theory of affine buildings is the Bruhat–Tits fixed-point theorem, which, in one of many formulations, states that, if $\Gamma$ is a group of isometries of an ...
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Palais's and Kobayashi's theorems on automorphism groups of geometric structures
My question concerns two results in the neighborhood of the standard theorem of Myers-Steenrod that isometry groups of Riemannian manifolds are Lie groups. Both appear in the first chapter of ...
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125
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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 ...
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What is an example of two Banach spaces $X,Y$ such that $X$ embeds isometrically but not linearly into $Y$?
By a result of Godefroy and Kalton if $X,Y$ are separable Banach spaces and $X$ embeds isometrically into $Y$, then $X$ embeds with a linear isometry into $Y$.
Is this result known to fail for ...
1
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Stochastic Integral + conditional expectation
Let $\overline{\widehat{Z}_i} = \frac{E_i\left[ \int_{t_i}^{t_{i+1}}\widehat{Z}_sds\right] }{\Delta t_i}$ with $\widehat{Z}$ a square integrable process, $\Delta t_i := t_{i+1} - t_i$, and $E_i$ ...
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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
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158
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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 $$\max\{\|x-p\...
- 577
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Isometries of fiber bundles
Let $F\to S\overset{\pi}{\to} B$ a Riemannian submersion with totally geodesic fibers.
Question: How much information about the isometries of $S$ we have if we know the isometries of $F$ and $B$? For ...
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Realizing mapping classes as isometries?
Let $\phi : M \to M$ be a diffeomorphism. Is there a metric $g$ on $M$ and a diffeomorphism $\psi$ isotopic to $\phi$ so that $\psi$ is an isometry with respect to $g$? I'm guessing the answer is no,...
- 5,299
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Every partial isometry extends
I am interested in metric spaces $X$ where every isometry between two subsets of the space extends to a full isometry $X \to X$. Is there a name for this kind of space? Is there some paper which ...
- 21
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Doubly-stochastic partial-isometric matrices
An $n\times n$ matrix $A$ with nonegative real entries $a_{ij}$ is said to be doubly stochastic if $\sum_{i=1}^na_{ij} = 1$,
for all $j$, and $\sum_{j=1}^na_{ij}=1$, for all $i$.
Much is known [1] ...
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Can a knotted sphere isometrically embed into $\mathbb R^3$?
All smooth simple closed curves in $\mathbb R^3$ (knotted or not) can be isometrically embedded into $\mathbb R^2$ as a circle of equal arclength.
The situation for knotted spheres seems more ...
- 449
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3D similarities and quaternions?
As is well-known, in dimension 2, a linear map $f : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ is a direct similarity if, once we identify $\mathbb{R}^2$ with $\mathbb{C}$, $f$ is of the form
$$\forall z \...
- 2,132
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Uniqueness of function with range $\mathbb{S}^2$ under a constraint
Assume $g,f\colon A\subset\mathbb{R}^M\rightarrow\mathbb{S}^2$ are two bijective functions defined on the set $A$. Now assume a constraint $C$: $\forall x,y\in A, \exists R\in SO(3)\colon Rf(x)=f(y)\...
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Isometric stratification preserves volume?
Let $K\subset \mathbb{R}^k$ be a non-empty compact subset let $f:K \to K$ be Lipschitz and surjective. If, moreover, $f$ is an isometry then clearly $f$ preserves the Lebesgue measure of $K$.
I ...
- 4,906
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Separable Banach spaces isometric to quotient of a Banach space
We know that every separable Banach space is isometrically isomorphic to a quotient space of $(\ell^1,\|.\|_1)$. We also know that the norm defined by $\|x\|=(\|x\|_1^2+\|x\|_2^2)^{1/2}$ for all $x\in ...
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Characterization of extrinsic distance prevserving embedding (see the definition given!) from low dimensional Euclidean spaces to high dimensions
P.S. I asked the question on MSE more than a week ago, but didn't get any desired answer, so asking here.
Let $m < n \in \mathbb{N}$. Let us equip $\mathbb{R}^m, \mathbb{R}^n $ with their ...
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How does this orthogonality follow from the map being an isometry?
This is a step of a proof in the book Variational Problems in Geometry by Seiki Nishikawa. I will ignore the background and change some of the statements and notations for simplicity.
Let $(M,g)$ ...
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Local isometry implies covering map: nonempty boundary case [closed]
The following theorem is well known in the literature:
Let $M$ and $N$ be riemannian manifolds and let $f : M \to N$ be a local isometry. If $M$ is complete and $N$ is connected, then $f$ is a ...
- 2,001
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1
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What are the applications of the Mazur-Ulam Theorem?
Every bijective isometry between normed spaces is affine. This well-known and beautiful statement, the Mazur-Ulam Theorem, was proved in 1932, but the proof has been simplified and polished in years, ...
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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 ...
- 2,087
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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 ...
- 445
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Pogorelov's rigidity theorem vs Cohn-Vossen rigidity theorem
There is the following rigidity theorem of Cohn-Vossen as stated on p. 86 of these lecture notes: http://www.math.brown.edu/~deigen/chern.pdf
Any isometry between two closed smooth convex surfaces (...
- 20.3k
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Cohn-Vossen rigidity theorem in hyperbolic space
There is the following rigidity theorem of Cohn-Vossen as stated on p. 86 of these lecture notes: http://www.math.brown.edu/~deigen/chern.pdf
Any isometry between two closed smooth convex surfaces in ...
- 20.3k
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Uniform distribution of points on Riemannian manifolds
Recently, I came across a beautiful paper by Arnol'd and Krylov (Uniform distribution of points on a sphere...) that contains the following theorem:
Theorem: Let A and B be two rotations of the ...
- 1,054
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0
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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 ...
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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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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 $R^{(n_k)}$ $(V_1, ..., V_k)$ ...
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If $E\oplus_\phi E \cong E\oplus_\psi E,$ does it imply that $\phi= \psi$?
Let $E\neq \{0\}$ be a Banach space.
For each $p\in[1,\infty), $ we define
$$E\oplus_p E = \{(x,y): x\in E, y\in E, \|(x,y)\| = \sqrt[p]{\|x\|^p + \|y\|^p}\}.$$
Let $F$ be another Banach space.
By $E\...
- 603
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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
$$r(p) = \min_{g \in \Gamma \setminus\{e\} } d(p, g(p)...
- 2,001
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Laplace-Beltrami and the isometry group
H$\vphantom{a}$i. Consider the Laplacian on $\mathbb R^n$,
$$
\Delta=\partial_i^2
$$
It is easy to prove that the most general differential operator that commutes with rotations and translations is ...
6
votes
2
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Are all symmetries of the Dirichlet functional isometries?
This is a cross-post from MSE (no answer there).
Let $M,N$ be oriented $d$-dimensional Riemannian manifolds, $M$ compact*, and let $f:M \to N$ be smooth.
Consider the Dirichlet energy functional: $...
- 6,499
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Conformal harmonic maps in high dimensions are scaled isometries
This is a cross-post from MSE (where I got no answer).
It is well-known that conformal maps between $2$-dimensional Riemannian manifolds are harmonic.
I discovered lately that in dimension $d>2$, ...
- 6,499
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2
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491
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Do curvature differences obstruct a.e orientation-preserving isometries?
Is there an example of a pair $M,N$ of connected, oriented equidimensional Riemannian manifolds with the following properties:
$M$ is everywhere non-flat, $N$ is flat.
There exist a map $f:M \to N$ ...
- 6,499
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1
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322
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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 ...
- 1,085
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2
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Norms on $\mathbb{R}^d$ whose linear isometries are the hypercube group
It is a known fact that for any $2\neq p\in[1,\infty]$, the linear isometries for the corresponding norm $\|\cdot\|_p$ on $\mathbb{R}^d$ is the set of all square-matrices with entries in $\{-1,1,0\}$, ...
- 2,410