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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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$C(K_1)\cong C(K_2)$ if and only if $K_1\cong K_2$

I had read the following statement in a book without proof. One of the directions is trivial, however the other is not: Let $C(K_j)$ the Banach space of continuous functions $f:K_j\to \mathbb{R}$ ...
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Can a finite Wasserstein metric on Euclidean support be embedded in a Euclidean space?

Thanks for everyone's help with understanding finite metric embeddings in Euclidean space. I have a follow-up question. Say we have the Wasserstein distance between $n$ distributions in Euclidean ...
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Riemannian Manifolds (Lee) Exercise 6-28 (d): Isometries converging pointwise converge topologically

Suppose $M$ is a connected, complete Riemannian manifold, $\mathrm{Iso}(M)$ is a smooth Lie group composed of all isometries of $M$, and $\phi_n$ is a sequence of isometries converging to an isometry $...
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Conditions on a finite metric that guarantees embedding in Euclidean space? [duplicate]

If we have $n$ points in some metric space, do there exist coordinates for the $n$ points in an $n-1$ dimensional Euclidean space with exactly the same pairwise distances as in the original space? ...
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A detail in the proof of Killing-Hopf theorem for Euclidean surface.

I am reading the book Geometry of Surfaces by Stillwell. In chapter $2$, he proves the following theorem: Theorem: (Killing-Hopf) Each complete, connected Euclidean surface is of the form $\mathbb{R}^...
Zoudelong's user avatar
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Spectral theorem for isometries

I will soon have an exam, and there is something that I simply don't understand: the spectral theorem for orthogonal matrices/endomorphisms (isometries). $\phi$ is an isometry $\Leftrightarrow$ There ...
metamathics's user avatar
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Is sequence $x_n = A^nBC^n x$ either bounded or distributed linearly?

Given 3 invertible matrices $A,B,C \in \mathbb R^{k\times k}$ and a starting vector $x \in \mathbb R^k$, $x\not= 0$, define a sequence $(x_n)$ by $x_n = A^nBC^n\cdot x$. Is it always the case that $\|...
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How to prove $\text{Sym}^+(H)$ is an abelian group.

Let $H$ be a shape, i.e., a non-empty compact set in $\mathbb{R}^2$. Denote by $\text{Sym}(H)$ the set of isometries preserving $H$ and by $\text{Sym}^+(H)$ the set of positive orientation-preserving ...
Alex Nguyen's user avatar
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an isometry preserving an equilateral triangle will preserve its vertices

Prove that an isometry preserving an equilateral triangle will preserve its vertices, i.e if $\Delta ABC$ is an equilateral triangle and $f$ is an isometry s.t $f(\Delta ABC)=\Delta ABC$ then $\{A,B,C\...
Alex Nguyen's user avatar
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Isometry group of quadric model of anti-de Sitter space

I am learning Lorentzian geometry on my own. Consider the space $\mathbb R^{p+2}$ endowed with the bilinear form $$\langle x, y \rangle_{p,2} = \sum_{i = 1}^{p+2} x_iy_i -x_{p+1}y_{p+1}-x_{p+2}y_{p+2}$...
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Decomposing a certain kind of isometry of modules

Let $R$ be a finite ring, $R_1 \times \dots \times R_m$ its decomposition into finite local rings. Let $\pi_i$ denote the projection to each factor $R_i$. Then $\pi_i$ extends componentwise to a map $\...
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If $f : U \rightarrow X$ is an isometric inclusion of Banach spaces, does $f' : X' \rightarrow U'$ have a bounded generalized inverse?

Let $X$ be a Banach space and let $U$ be a closed Banach subspace. The inclusion mapping $$ f : U \rightarrow X $$ induces a dual mapping $$ f' : X' \rightarrow U' $$ I am wondering about the ...
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What is extension of an isometry?

I am reading Lorentzian geometry. I found the following definition. Definition: A Lorentzian manifold $M$ has maximal isometry group if the action of $\text{Isom(M)}$ is transitive and, for every ...
yyffds's user avatar
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Isometries in $\mathbb{R}^{3}$

I don´t know if my solution to this problem is complete and 100% correct. I´d appreciate some feedback. Let $f$ and $h$ be two isometries in $(\mathbb{R}^{3}, \langle, \rangle)$. If $f$ is the ...
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Prove isometric embedding of $S^2$ into $\mathbf R^5$

I'm trying to solve Exercise $132$ on the last page of this pdf Let $S^2$ denote the unit sphere. The map $f: \mathbf{R}^3 \rightarrow \mathbf{R}^5$ is given by $$ f(x, y, z)=\left(y z, z x, x y, \...
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Is every isometry invariant measure on Euclidean space equivalent to Hausdorff measure?

One of the properties of Hausdorff measure on $\mathbb{R}^d$ is invariance under isometry. I wonder that if every isometry-invariant measure is similar to Hausdorff measure. To be rigorous, let $\mu$ ...
Daeyoung Jeong's user avatar
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Tangent developable is locally isometric to an open set in ℝ²

I'm trying to solve this problem about tangent developable from a differential geometry exam $\sf 1996\ Q3$: My working: First part $\ldots\ldots$deduce that $Σ$ is ruled (that is, each point of $Σ$ ...
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$V$ Euclidean iff there exist linear isometry that $f(v)=w$ for $|v|=|w|$

From the book 'a course in metric geometry' exercise 1.2.24 : Let $V$ be a finite-dimensional normed space. prove that V is Euclidean iff for any tow vectors $v,w\in V$ such that $|v|=|w|$ there ...
hr1380's user avatar
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Are there self-adjoint operators with eigenvalues 1, -1 that are not isometries?

Let $f$ be a self-adjoint operator in an euclidean metric vector space $(V, g)$. a) Prove that the following are equivalent. i) The only eigenvalues of $f$ are 0, 1. ii) $f$ is an orthogonal ...
MrGran's user avatar
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isometries as products of reflections [closed]

I am taking a course on geometric transformations in the 2-dimensional Euclidean plane. I have been told that all isometries (length-preserving transformations) are products of reflections about ...
mathman's user avatar
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Let $(M,g)$ be a (compact) Riemannian manifold with isometry group $G$. Who is the isometry group of the volume form $\rho d\operatorname{vol}$?

Let $(M,g)$ be a (compact) Riemannian manifold with isometry group $G$. That is, if $F\in G$, $F^*d\operatorname{vol}=d\operatorname{vol}$. Who is the isometry group of the volume form $\rho d\...
Gomes93's user avatar
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How is the real line a geodesic in the complex upper half-plane?

I'm in the middle of trying to prove that all hyperbolic isometries are Möbius transformations in the upper half-plane and I keep seeing people mention looking at points on the real line and ...
sympie's user avatar
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isometries and unitary operators, Specht theorem

I was looking at the properties of the trace operator $\operatorname{tr}$, in particular the properties of the trace regarding isometric conjugation. We say that $T\in \mathcal{H}$ is an isometry if $...
ana's user avatar
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Extending linear isometry from subspace to whole space

Problem: Let $(V, \langle \cdot, \cdot \rangle_V)$ and $(W, \langle \cdot, \cdot \rangle_W)$ be finite-dimensional inner-product spaces, such that $\dim V\leq \dim W$, and let $U\subset V$ be a ...
categoricallystupid's user avatar
2 votes
1 answer
53 views

${\rm Hess}~r$ is scalar matrix $\implies$ $M$ is isometric to the space form

I'm trying to prove the rigidity part of a theorem in my paper, which requires the use of the classical Hessian comparison theorem's rigidity part: $${\rm Hess}~r=\frac{{\rm sn}_k'}{{\rm sn}_k}{\rm d}...
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$C^1$ and affine map

Let $f:\mathbb R^n \to \mathbb R^n$ be a $C^1$ length-preserving map (i.e., preserving the length of curves). Then we have that $Df \in O(n)$ for every $x \in \mathbb R^n$. It follows that $f$ is ...
Frederick Manfred's user avatar
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1 answer
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How does $X = Y \oplus U = Y \oplus V$ for a Banach space $X$ and some subspaces imply that $U$ isometric $V$?

I'm trying to prove the following: Let $X$ be a Banach space and let $Y,U,V \subseteq X$ be subspaces such that $X = Y \oplus U = Y \oplus V$. Then $U$ and $V$ are isometrically isomorphic. So far I ...
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3 answers
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Matrix function derivative. Introduction

The author of this question was close to determining the derivative of the function of dual variable, when we consider matrices isomorphic (algebraically and topologically) to dual numbers: $$(a+\...
Иван Петров's user avatar
1 vote
1 answer
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Veryshynin HDP Exercise 4.1.4: Equivalent definitions of isometry

I'm trying to show the following: Let $A$ be an $m \times n$ matrix with $m\geq n$. Prove that the following statements are equivalent. $A^{\top} A = I_n$. $P = A A^{\top} $ is an orthogonal ...
pbb's user avatar
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Does a map that preserves distance imply isometry?

It feels intuitive for this to be true. Let us say I have a chart from $\mathbb{R^2}$ to some section of a manifold in $\mathbb{R^3}$. If the distance in $\mathbb{R^2}$ between two points is equal to ...
Dr. Ernesto Chinchilla's user avatar
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1 answer
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Isometry between cone and cylinder [closed]

In a certain exercise I have been asked to find an isometry between a portion of the cylinder $S = \{ x^2+y^2 = 2: 0 < z < 1\}$ and the complete cone $S_* = \{x^2+y^2 = 2z^2: 0 < z < 1 \}$...
Emmy N.'s user avatar
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Isometry between triangle and cone

In a certain exercise I have been asked to say if there is an isometry between the triangle $T =\{ z = 0, 0 < x, y, x + y < 1 \}$ and the cone $C = \{x^2 + y^2 =\frac{z^2}{4}, 0 < z < 2\}$....
Emmy N.'s user avatar
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1 vote
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Prove $\ell^1 \cong c_0*$

To prove: $T:l^1 \to c_0^*, \langle Tx, y\rangle = \sum x_ky_k$ is an isometric isomorphism Is this proof correct? Particularly, I dont know if I proved injectivity correctly Proof: (i) linearity: let ...
juan19.99's user avatar
1 vote
0 answers
72 views

The isometry group of a metric space with the topology of pointwise convergence is a topological group

Let $(X,d)$ be a metric space and let $\mathrm{Iso}(X)$ denote the group of isometries of $(X,d)$, where the group operation is the composition. Equip $\mathrm{Iso}(X)$ with the topology of pointwise ...
jenda358's user avatar
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0 answers
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Showing an abstract surface (with a specific metric) is isometric to a helicoid

$\DeclareMathOperator{\sech}{sech}$I've been trying to understand a question in Barrett O'Neill's Elementary Differential Geometry that goes as follows: "Show that the geometric surface in ...
MathMetalMegaman's user avatar
1 vote
1 answer
75 views

Prove that a change-of-basis map is an isometry between complex hilbert spaces, to prove uniqueness of purifications

I am trying to prove the following For this I should use the following fact: Let $ρ_A = \sum_{i=1}^r p_i|e_i⟩⟨e_i|$, where $p_i$ are the nonzero eigenvalues of $ρ_A$ and $|e_i⟩$ corresponding ...
some_math_guy's user avatar
3 votes
1 answer
255 views

Is a map that preserves torsion and curvature of all smooth curves an isometry?

In my introductory differential geometry class, we learnt that isometries preserve the torsion $\tau$ and curvature $\kappa$ of curves. I was wondering if the converse of this statement is true: if a ...
Nathan Hart-Hodgson's user avatar
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0 answers
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Which groups are achievable as achievable as the scale factors of a manifold?

Let $(M, g)$ be a Riemannian manifold. I say $\lambda\in\mathbb R^\times$ is a scale factor of $M$ if $(M, g)$ is isometric to $(M, \lambda g)$. Composing isometries multiplies the scale factor, so ...
Derivative's user avatar
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5 votes
1 answer
247 views

When is isometry space of compact metric space a manifold?

Supposse we have a compact metric space $(X, d)$. Let $(\text{Iso}(X, d), D)$ be a metric space of all isometries $f : X \rightarrow X$ with a metric $D$ defined with: $$D(f, g) = \text{sup}\{d(f(x), ...
Goki's user avatar
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1 vote
0 answers
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Isometry of surface

Let $S$ be a set of points $ (x,y,z)\in\mathbb{R^3}$ that satisfy the equation $x^3+y^2+z^2=1$, $p_1=(0,1,0)$ and $p_2=(0,0,1)$ points in $S$ with the same gaussian curvature,prove that $f: S \to S$ ...
Andreadel1988's user avatar
1 vote
1 answer
109 views

Isometric Isomorphism From $\mathbb{C}^n \oplus M([a,b])$ Onto $(C^n[a,b])^*$

I have the following norm in $C^n[a,b]$: $$\| f \| = \sum_{k=0}^{n-1}|f^{(k)}(a)| + \sup_{[a,b]} |f^{(n)}|$$ I want to show that: $$(\lambda_0,...,\lambda_{n-1},\mu)\mapsto \left( f\mapsto\sum_{k=0}^{...
CauchyChaos's user avatar
1 vote
1 answer
52 views

Transform $3D$ grid to dimetric projection [closed]

I'm rendering rhombus textures in a $3D$ world: Here are the rhombus textures rendered in a grid in perspective projection with $FOV = 45^\circ$: Here is the same grid scene, but in dimetric ...
user7401478's user avatar
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0 answers
51 views

Isometries of a submanifold with induced metric

I need help to find isometries of a submanifold of a semi-Riemannian manifold. To be crystal clear, let me start with what I mean by an isometry. $\textbf{Definition:}$ Let $(M,g)$ be a semi-...
lolabol's user avatar
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1 vote
1 answer
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Showing that this embedding is isometric

Let $X$ be a metric space and $B(X)$ be the space of bounded functions $f: X \to \mathbb{C}$. Consider the embedding $J : X \to B(X)$ defined via $y \mapsto J(x)(y) = d(x, y) − d(x_0, y)$ for some ...
MathGeek's user avatar
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2 votes
1 answer
98 views

If $A$ and $B$ are countable dense subsets of compact metric space $(X, d)$ with infinitely many isometries, is there isometry such that $f(a) \in B$.

Let $(X, d)$ be a separable and compact metric space with infinitely many isometires. Let $A, B$ be infinitely countable dense subsets of $X$. Is there an isometry $f : X \rightarrow X$ such that $f(a)...
Goki's user avatar
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0 answers
191 views

When is a Riemannian metric the pullback of the Euclidean metric?

Let $F:M \to (N,\bar{g})$ be a smooth map between two smooth manifolds $M \subset \mathbb{R}^2$ and $N \subset \mathbb{R}^2$ of dimension 2, with $\bar{g}$ the Euclidean metric. From what I ...
arthur's user avatar
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1 answer
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Isometry from $\mathbb{C}^2$ to $\mathbb{C}^3$?

I've been interested in isometries of complex Hilbert spaces recently and wondered if it's possible to embed $\mathbb{C}^n$ isometrically into $\mathbb{C}^{n+1}$. Rather than tackling the general ...
Henrymerrild's user avatar
-1 votes
1 answer
73 views

Is isometric to $\mathbb R^n$ implies vector space?

Suppose $(E,d_E)$ is a metric space that is isometric to $\mathbb R^3$ with Euclidean distance. I have some impedent : (1) I want to state that $E$ is a metric space of dimension $n$, but as far as I ...
PermQi's user avatar
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1 vote
1 answer
160 views

If 2 vector space are isometric then they are isomorphic

We call a map between 2 normed vector spaces is isometric isomorphism if it is an isometry and an linear isomorphism. Then, these 2 vector spaces are called isometrically isomorphic. Suppose $X,\,Y$ ...
PermQi's user avatar
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0 answers
54 views

isometric isomorphism implies diffeomorphism

Suppose $E$ be a $\mathbb R-$vector space that is isometrically isomorphic to $\mathbb R^n$ via the map $f:\, E\to\mathbb R^n$, which means $f$ is bijective and \begin{align} d_E(x,y)&=d_{\mathbb ...
PermQi's user avatar
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