Questions tagged [isometry]

An isometry is a map between metric spaces that preserves the distance. This tag is for questions relating to isometries.

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66 views

Existence of isometry into $\mathbb{R}^n$

Let us define the following : the pseudo-power set $P = \{A \subseteq \mathbb{Z} \text{ : }A \text{ is a finite set}\}$ the function $d : P \times P \to \mathbb{R}$ as $d(A,B) = \text{cardinality of ...
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42 views

Brezis exercise 3.20

I have the following question. Let $E$ be a Banach space and separable, then exists an isometry form $E$ in $\ell^{\infty}$. The hint is: Consider Since $K = B_{E^{\star}}$ is compac and metrizable ...
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Lower bound for isometric immersions

I just read Azov's article in the considered two classes of Riemannian metrics, \begin{align*} ds^2&=du_1^2+f(u_1)\sum_{i=2}^ldu_i,&f>0\\ ds^2&=g^2(u_1)\sum_{i=2}^ldu_i^2 ,&g>0\...
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59 views

Can a symmetric, open, simply connected set in $\mathbb{R}^2$ be partitioned into two congruent sets?

Can an open, "symmetric" (in a non-rigorous sense), simply connected set $S$ in $\mathbb{R}^2$ be partitioned into two congruent sets? This sounds obvious (eg. for a circle, cut it in half) ...
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Finding isometries

I was wondering if for a given set of points $x_1,...,x_n$, $n > 2$, and two metric spaces $(X,d_1),(Y,d_2)$, $X \neq Y$, it s possible to find a continuous mapping $T: X \rightarrow Y $, such ...
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34 views

About isometries in $\mathbb{R}^3$ [closed]

I know that in $\mathbb{R}^2$ if $m$ is an orientation-reversing motion then $m^2$ is a translation. But what about $\mathbb{R}^3$? If $m$ is an orientation-reversing motion in $\mathbb{R}^3$ then is $...
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22 views

If an isometry $f$ of a length space $X$ fixes a ball $B(x,r)$ for some $x$ and $r>0$, is it true that $f$ is the identity map?

Let $(X,d,L)$ be a length space, that is, $d = \inf L$, where the inf is taken over all curves with fixed endpoints. In $\mathbb{R}^2$ we have a results that says: if an isometry fixes 3 points, then ...
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Are $l^p$ and $(l^q)'$ canonically isomorphic or just isomorphic?

As we know from basic functional analysis, the sequence spaces $l^p$ and $(l^q)'$ are isometrically isomorphic. Are they considered to be canonically isomorphic or just isomorphic? Based on the ...
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Can we transform a metric space to be isomorphic [closed]

Suppose we know that there's a metric subspace that is not isomorphic/isometric to another. Can it be transformed such that it becomes isometric to the other? If so, how can it be done? Can the same ...
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Are principal curvatures intrinsic when both are nonzero?

If we start with a surface in $\mathbb{R}^3$ with nonzero principal curvatures $\kappa_1,\kappa_2\neq 0$ is it ever possible to choose an isometric embedding of that surface in $\mathbb{R}^3$ with ...
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$\mathrm{Iso}(X)$ is locally compact if $X$ is locally compact

Let $(X, d)$ be a locally compact metric space. Define $\mathrm{Iso}(X)$, the isometry group of $X$, as the set of all surjective isometries of $X$. Is it true that $\mathrm{Iso}(X)$ is locally ...
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24 views

Strongly equivalence and isometric spaces

I have the following definitions Definition; Two metric spaces $(X,d_1)$ and $(X,d_2)$ are said to be isometric if there is bijective isometry from one space to other. Definition:Two metric spaces $(X,...
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When a local isometric immersion become global?

I have a question that I have not been able to answer clearly. When a local isometric immersion become global? For example, I know the following result, if $\phi:M\to N$ be a differentiable mapping ...
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Probability density function (pdf) of $\|{\bf X}{\bf a}\|_{2}^{2}$, given the pdf of the elements of random matrix ${\bf X}$

I am reading some papers regarding Johnson-Lindenstrauss lemma and some proofs of the restricted isometry property (RIP) in compressive sensing. I have the following problem: ${\bf X} \in \mathbb{C}^{...
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2answers
104 views

Local isometric immersion from $\mathbb H^{n}$ into $\mathbb R^{2n-1}$

I found this local isometric immersion from $\mathbb H^{n}$ into $\mathbb R^{2n-1}$, given by Schur (1886) en Über die Deformation der Räume constanten Riemannschen Krümmungsmaasses as follows, $(1\...
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Using inner product to prove that a function is an isometry...

$\textbf{My problem:}$ Let $f:\mathbb{R^m}\rightarrow \mathbb{R^m}$ be a function of class $C^1$ such that for each $x\in \mathbb{R^m}$, $|f'(x)\cdot v|=|v|$. Prove that $|f(x)-f(y)|=|x-y|$ for each $...
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Is every isometric relation derivable and constructible (from/to some set of points) via rigid transformations? Can these be expressed exhaustively?

There are three types of rigid transformations that can be combined to create an isometry: reflection, translation, rotation. The former two can be applied to any function to create a new (or ...
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Proving $A$ is an isometry on $\mathbb{R}^3$ if $A = I - 2vv^T$

Let $v$ be a vector of unit length in $\mathbb{R}^3$, and $A = I - 2vv^T$. Prove that $A$ is an isometry. I need to prove that $A$ preserves dot products, i.e., $A x \cdot Ay = x \cdot y$ for any $x,...
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Prove that $\det R = \pm 1$ iff $R \in \operatorname{Isom}(V)$, where $V$ is a finite dimensional real inner product space [duplicate]

Let $V$ be an $n$-dimensional real inner product space. An isometry on $V$ is an operator $R$ with $\langle Rx,Ry \rangle$ for all $x,y \in V$. The determinant of an Endomorphism can be defined i many ...
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31 views

Deciding isomorphism of two sets of points given the distances between points of the same set

Consider a set of $m$ points in $\mathbb{R}^n$, $2 \le m \le {n \choose 2}$. We do not know the coordinates of the points, but we know the distances of each point from any other point. However, for ...
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Discrete groups acting on hyperbolic space

I am interested in groups of isometries which act discretely on uniform spaces of constant curvature, ie $\mathbb{S}^n$, $\mathbb{E}^n$, and $\mathbb{H}^n$. I believe I am right that any such group ...
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quadratic variation and local martingale

Consider a predictable process $H_t$ and a continuous local martingale $M_t$ for $t \in [0,T]$ I want to show that $$X_t:=\left(\int_0^t H_s dM_s\right)^2 - \int_0^t H^2_s d \langle M_s\rangle $$ is ...
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162 views

Visualize the isometry of the sphere $S^2$ is $S^3/(\mathbf{Z}/2) ?$

How to visualize the rotational symmetry group or isometry of the sphere $S^2$ is $$S^3/(\mathbf{Z}/2) ?$$ I meant that because that the rotational symmetry group is $SO(3)=RP^3=S^3/(\mathbf{Z}/2) $. ...
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47 views

Does Euler's rotation theorem assume a continuous movement over time?

Or can we apply ANY isometry in a sphere and it will have a fixed diameter? Regardless if it is discontinuous in time. Would absolutely any discontinuous isometry be a rotation in a sphere?
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27 views

Isometric Embeddings into Hyperbolic Space

When can a finite metric space $X$ be isometrically embedded into Hyperbolic space $\mathbb{H}^{n}$? It seems there are results for isometric embeddings into $\mathbb{R}^{n}$ Maehara 2013 with ...
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40 views

The isometry group of SL(2,C)

I want to find the isometry group of $SL(2,C)$ and after that I want to find the isotropy group $H$, ( $H$ is the stabilizer of an element by using the action of the isometry group on $SL(2,C)$ )
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Determining the type of isometry

I am working on a problem that says: Classify and discuss the isometry $F: \mathbb{E}^3 \to \mathbb{E}^3: (p_1,p_2,p_3)\mapsto (p_2-2,p_3+1,8-p_1)$. I really don't know where to begin on problems like ...
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38 views

Idea about isometric embedding in two dimension

I was thinking how to embed a Riemannian manifold in the Euclidean space. I had an idea, then I found the Nash embedding theorem but I was expecting something different, this is what I thought: ...
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2answers
58 views

Shape whose group of isometries has order 7 [closed]

I have recently been tasked with drawing a shape whose symmetry group of isometries have an order of 7. Though I am unsure of how to go about this as I have only ever drawn shapes whose symmetry ...
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35 views

Does a purely Euclidean proof that an orientation preserving isometry is either a translation or a rotation about some point, exist?

In his book ALGEBRA, $2^{\text{nd}}$ ed., Artin proves (Ch. 6 - Symmetry, Lemma 6.3.5) that an orientation preserving isometry $f$ that has the form $m=t_a \rho_\theta$ with $\theta \neq0$, is a ...
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1answer
30 views

Isometric map vs symmetry transformation in a finite-dimensional Hilbert space

For a given finite-dimensional complex Hilbert space $\mathcal{H}$ with the inner product $\langle \cdot | \cdot\rangle$, a map $f: \mathcal{H} \to \mathcal{H}$ is said to be an isometric map if $$ ||...
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1answer
30 views

Alternating forms and isometries of the underlying vector space

Does a linear isometry of $\mathbb{R}^n$ yields a linear isometry of $\mathbb{R}^n \wedge \mathbb{R}^n$ ? If yes, why ? How to prove it elegantly, without explicit coordinate computations ? Using, for ...
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0answers
61 views

Isometry groups of $\mathbb H^n(R)$ (hyperbolic spaces)

I know that the group of isometries on $\mathbb H^n(R)$ is $O^+(n,1)$ (the orthochronous lorentz group) which i have proved using the Hyperboloid model. I have also been able to show for n=2 case ...
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3answers
138 views

All Hilbert spaces are isometric to $l^2(E)$ - how?

I was reading about Hilbert spaces and came across this line on Wikipedia: By choosing a Hilbert basis (i.e., a maximal orthonormal subset of $L^2$ or any Hilbert space), one sees that all Hilbert ...
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1answer
66 views

Pointwise equality of codimension-zero immersions

Suppose $f,g: U \subset \mathbb{R}^n \to \mathbb{R}^n$ are smooth immersions from a closed and bounded domain $U$ such that $f|_{\partial U} = g|_{\partial U}$. If $f$ and $g$ induce the same ...
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Do Spherical Lenses Have Circle Symmetry?

For the purposes of this question, a spherical lens is the interesection of two spherical half-spaces in $\mathbb{S}^3$. (A spherical half-space is the intersection of $\mathbb{S}^3$ with a half-space ...
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1answer
31 views

Interpreting the isometries of a tetrahedron

A tetrahedron has $12$ rotational symmetries and $24$ isometries in total. This means that the group of isometries is isomorphic to $S_4.$ If we denote the vertices of a tetrahedron as $1$, $2$, $3$, ...
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1answer
105 views

Not every finite metric space embeds in an $\mathbb{R}^{k}$

This problem is present in "Supplements to the Exercises in Chapter 1-7 of Walter Rudin's Principles of Mathematical analysis" by Prof. George M. Bergman, which states as follows, Let $X$ be ...
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Proportionality of normal curvatures in bending of two identical surfaces

In isometric mapping or continuous bending deformations between two surfaces with same constant Gauss curvature $$ K_{12}= K_{34} $$ of constant principal normal curvature products: $$ K_{12}=k_1 k_2,...
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45 views

find all isometries that commute with this linear application

as in the title given the application $F(x_1,x_2,x_3)=(-\frac{1}{2}x_1-\frac{\sqrt3}{2}x_2+\frac{1}{2};-\frac{\sqrt3}{2}x_1+\frac{1}{2}x_2+\frac{\sqrt3}{6};-x_3)$ 1)find all isometries of $\mathbb R^...
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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 ...
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33 views

Proving that $J : \varphi \longmapsto J_{\varphi}$ is an isometric isomorphism of the Banach algebra $L_{\infty} (X,E).$

Let $(X,\mathcal A, E, \mathcal H)$ be a spectral measure space. Then show that the map $J : \varphi \longmapsto J_{\varphi} = \int_X \varphi\ dE$ is an isometric isomorphism of the Banach algebra $L_{...
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33 views

In search of word to describe transformation by 'orthogonal' functions (e.g. donut torus to flat torus).

I am looking for the mathematical name of transformations between manifolds that consist of sets of functions whose gradients are orthogonal. For example, the almost everywhere valid mapping of the ...
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72 views

Proving what the isometry group of a "chess board" is

"Find the group of symmetries of a chess board", was the original question. As stated is not very rigorous, i intrepreted it as asking to find a (setwise) stabilizer of the group of $\mathbb{...
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1answer
54 views

Let $M = {\{x \in l^p : x(2n) = 0 \ \text{for all n}}\}$, $1 \le p \le \infty$. Show that $l^p/M$ is isometrically isomorphic to $l^p$.

The following is Exercise 6 page 72 in Functional Analysis book of Conway : Let $M = {\{x \in l^p : x(2n) = 0 \ \text{for all n}}\}$, $1 \le p \le \infty$. Show that $l^p/M$ is isometrically ...
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36 views

Proving that linear isometries preserve angles without the polarization identity

Let $V$ be an inner product space, and let $T : V \to V$ be a linear isometry, that is $$ \|Tv\| = \|v\| \quad \text{for all $v \in V$}. $$ I would like to prove that $T$ also preserves inner products,...
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41 views

Bending to Catenoid without twist

Minimum radius $r=c$ central line of a Catenoid $$ \sqrt{x^2+y^2} =c \cosh (z/c)$$ is to be mapped by isometrically bending it to a straight line (black) of length $ 2 \pi c $ without twist. With ...
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1answer
18 views

Isometry between two line segments

Let $T$ be a subspace of the plane formed by two unitary line segments: one horizontal line,$I$, and one vertical line, $J$, which the origin is the midpoint of $I$. Show that exist an isometry $f:I \...
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32 views

Proving equivalence of different definitions of isometry

In my lecture on differential geometry, we introduced the following definition: Definition 1: Let $\phi : M \longrightarrow N$ be a smooth map between regular surfaces. Then the map $\phi$ is called ...
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1answer
56 views

local isometries with same derivative on Riemannian manifolds

Let $(M_1,g_1)$ and $(M_2,g_2)$ be two connected Riemannian manifolds and let $f_1$ and $f_2$ be two local isometries such that $f_1(p)=f_1(p)$ and $f_{1*}(v)=f_{2*}(v)$ for some $p\in M$ and $\forall ...

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