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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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How to show isometry of the space through plane?

I am totally new to the isometries of the plane and space. I have to prove that the map $$R(\vec{x})=\vec{x}-2(\vec{x}\bullet\hat{n})\hat{n}$$ from $R^{3}$ to $R^{3}$ is isometry of the space $R^{3}.$ ...
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Isometries of the Plane, Euclidean space $R^3$ and isometries of the Platonic polyhedra.

I want to study the isometries of plane, Euclidean space, and the platonic polyhedra. I am new to this topics. Can any one suggest books that contain these topic with details and basic explanation ...
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Differential Geometry - Local Isometries

Consider the following surfaces in $ \mathbb R ^3 $: $$ \Sigma _ 1 = \{ (x_1, x_2, x_3) \in \mathbb R ^3 : x_3 = x_1x_2 \} \\ \Sigma_2 = \{ (x_1, x_2, x_3) \in \mathbb R ^3 : x_3 = \dfrac{x_1^2-...
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prove $\dim(\operatorname{range}(T)) = \dim(\operatorname{range}(\sqrt{T^*T}))$

I'm a student and I'm studying linear algebra. in Polar Decomposition we have: for a linear operator $T$, there exist a linear isometry $S$ that: $$ T =S\sqrt{T^*T}$$ so if $S$ is a linear ...
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If two spheres are isometric, does there exist a bijective isometry $T:S\to S$ with $\|Tu-\alpha Tv\|_Y \leq \|u-\alpha v\|_X$ for all $\alpha>0?$

Let $$(S,\|\cdot\|) = \{(x,y)\in \mathbb{R}^2: \|(x,y)\| =1\},$$ that is, $S$ is the collection of all norm one vectors in $\mathbb{R}^2$ with respect to the norm $\|\cdot\|.$ Question: Let $\|\...
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1answer
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Subgroup of Möbius transformations which are isometries with respect to the standard metric on the Riemann sphere

I'm trying to find which subgroup of Mobius transformations are isometries with respect to the standard metric on the Riemann sphere (the one induced from the Euclidean metric on $\mathbb{R}^3$). The ...
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What is the set of all isometric matrix in $\mathbb{R}^{k \times d}$?

An isometry from metric space $X=\mathbb{R}^{d}$ to metric space $Y=\mathbb{R}^{k}$ with usual norm for both spaces is the following: $$ \Phi: \mathbb{R}^{d} \rightarrow \mathbb{R}^{k} $$ where $\Phi(...
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Map preserves angle iff scalar multiple of isometry

How do I prove that a map preserves the angle if and only if it's the scalar multiple of an isometry. I get the "if" direction by using definition of isometry. How do I show the other direction, i.e....
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Find the number of isometries of the brick.

Consider a brick of length $3 cm$ ,breadth $=2$ cm and height $=1$ cm Find the number of isometries of the brick. I know that isometry means distance preserving metrics.But how to find the ...
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1answer
29 views

Isometries on a Banach space converging pointwise

I'm trying to find a Banach space $V$ with closed unit ball $B$ and a sequence of isometries $(f_n:V\to V)$ such that $(f_n)$ converges pointwise in $B$ but not uniformly in $B$. My first attempts ...
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Inversion as hyperbolic isometry (Poincaré disk model).

I'm beginning to study hyperbolic geometry in the Poincaré disk model, which is described as $$D = \{z \in \mathbb{C} : |z|<1 \},$$ and I need to show that inversion about a circle orthogonal to $\...
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1answer
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What is an example of two non isometric closed balls of same radius?

Both balls need to be in one metric space. I come up with idea of metric space $X ={a,b,c}$ and $d(a,b) = d(a,c) = 2$, $d(b,c) = 1, d(x,x) = 0$. Then i can say that $B(a,1)$ and $B(b,1)$ are non ...
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1answer
63 views

Why the isometry group is not the orthogonal group?

I found the following result: If $V$ is a euclidean vector space of finite dimension $n$ and $B$ is an orthonormal basis of $V$, then an endomorphism $f:V\to V$ is an isometry iff its matrix ...
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1answer
52 views

Surjective differentiable map is an isometry

This is exercise 1.2 in Svetlana Katok's Fuchsian Groups. $\mathbb{H}$ is the upper half plane (with the hyperbolic metric), and $f:\mathbb{H}\rightarrow\mathbb{H}$ is a surjective $C^1$ map. I want ...
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Isometric embedding probability distributions with tree transportation cost into $\ell_1$

I was trying to solve the following problem, but don't quite know how to get started with working with the transportation metric. Let $T = ([n], E)$ be an unweighted, undirected tree with root $r \...
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1answer
22 views

How to show that there are exactly two motions mapping $P\to P'$ and $Q\to Q'$ given that $d(P,Q)=d(P',Q')$

Given points $P,Q,P',Q'\in\mathbb{E}^2$ with $d(P,Q)=d(P',Q')$. Show there exists exactly two motions (isometries) $T_{1},T_{2}$ that maps $P\to P'$ and $Q\to Q'$ How exactly would I attempt proving ...
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1answer
46 views

Which properties preserve by isometry?

I know that an isometry between two surfaces preserves the 1st fundamental coefficients, geodesic curvature, and Gaussian curvature. I wonder that how about the curvature and torsion of a curve on ...
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1answer
16 views

Three points fixed by the composition of an two isometries

I am in the final step of a proof on classifying the symmetries of $\mathbb{R}^2$. Suppose we have some symmetry $\sigma$ that fixes at least two points, say $A$ and $B$. Then consider $C$ which ...
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1answer
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Any symmetry that fixes three non-collinear points is the identity

I am asked to finish the following sentence: Let $\sigma$ be an isometry on $\mathbb{R}^2$, suppose it fixes the points $A$ and $B$ Suppose $\sigma$ also fixes a third point $C$ which is not on the ...
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1answer
44 views

Show that if x and y are of same length, there must exist an isometry with f(x) = y

Assume $x, y \in V$ with $V$ being an euclidean vector space. I have to show: There exists an isometry $f: V \rightarrow V$ with $f(x) = y$, if and only if $\left|\left|x\right|\right| = \left|\...
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1answer
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Iteration of a parabolic transformation

Let $a$ be a point of $S^{n-1}$ fixed by a parabolic transformation $\phi$ of $B^{n}$ (conformal ball model). One has to show that if $x$ is in $\bar{B^{n}}$ , then $$lim_{m \rightarrow \infty} \phi^...
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Isometry on Riemannian manifolds problem 3

I'm reading K&N Foundations of differential geom. vol 1, and at the page 169 is this theorem. And I'm asking, is the converse true?
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1answer
182 views

Are inner product-preserving maps always linear?

Let $E,F$ be Pre-Hilbert spaces and $T: E \rightarrow F$ be a map that preserves the inner product, that is $\langle Tu , Tv \rangle = \langle u , v \rangle$ for all $u,v \in E$. Must it be true that $...
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Isometries in Minkowski space

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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1answer
127 views

Exists a discrete Euclidean isometry group that's not finitely generated?

This is a variation on my other question. A hyperbolic isometry group may be discrete and not finitely generated. What about the Euclidean case? The generating set $S$ may contain an infinite number ...
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1answer
81 views

$T^{\ast}T$ unitary then $T$ isometry

An operator $T$ is an isometry if $||Tf||=||f||$ for all $f$ in a Hilbert space $H$. (a) Show that if $T$ is an isometry, then $<Tf,Tg>=<f,g>$ for every $f,g \in H$. Prove as result that $...
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1answer
41 views

Isometric embedding of $l_\infty$ to $L_\infty[0;1]$

I'm trying to find an isometric embedding of $l_\infty$ to $L_\infty[0;1]$, i.e such bounded operator $\textsf{T}: l_\infty \to H \subset L_\infty[0;1]$, that $||\textsf{T}(x)||_{L_\infty[0;1]}=||x||...
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isometries of angles

Find an isometry that maps ∠(1,1)(3,2)(2,2) to∠(3,−1)(3,2)(4,0) This is a problem for school but I don't just want the answer I want to know how to understand it. I graphed it on wolframalpha. I can ...
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An isometry between totally geodesic submanifolds is smooth?

I am studying Sharafutdinov's Convex sets in a manifold on nonnegative curvature. I have found the following statement, $S_0$ being a totally geodesic submanifold of an open Riemannian variety $M$ ...
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bijective isometry between hyperbolic spaces

Let $\alpha : V \rightarrow W$ be a vector space isomorphism and $\alpha^*:W^* \rightarrow V^*$ the dual isomorphism. Let $h_v:(V\oplus V^*) \times (V\oplus V^*) \rightarrow K, $$\space ((x,f),(y,g)) \...
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1answer
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Exists a discrete isometry group that's not finitely generated?

I'm considering the possible combinations of these properties for an isometry group: "finite", "finitely generated", "discrete". Obviously, a finite group is necessarily finitely generated and ...
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How to transform points from a plane to a different plane?

I have a plane $\Pi_1$ expressed as $ax+by+cy+d=0$ and a different plane $\Pi_2$ expressed as $ex+fy+gz+h=0$. I am looking for a transformation which rigidly brings points lying on $\Pi_1$ to points ...
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Question from Axler's Linear Algebra Done Right Regarding Isometries and Normal Operators

Exercise 7.C.11 reads: "Suppose $T_1,T_2$ are normal operators on $\mathcal{L}(\mathbb{F}^3)$ and both operators have $2,5,7$ as eigenvalues. Prove that there exists an isometry $S\in\mathcal{L}(\...
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Isometric Triangles in Hyperbolic Plane

If $\Delta_{1}$ and $\Delta_{2}$ are two geodesic triangles in $\mathbb{H}_{2}$ with the same interior angles $\alpha$ , $\beta$ and $\gamma$, then there is an isometry of $\mathbb{H}_{2}$ that takes ...
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2answers
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Constant Gauss curvature $\Rightarrow$ homogeneous?

Let $S\subset \mathbb{R}^3$ be an embedded surface and $g_S$ the induced metric from $\mathbb{R}^3$ onto $S$. Since isometries preserve Gaussian curvature, $S$ homogeneous $\Rightarrow S$ has constant ...
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1answer
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Image of basis under rotation

I am intending to learn Lie Algebras by following the online course on MIT OCW webpage. In the preface section of the class, the following is stated in the lecture notes: "Consider the set of ...
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Is my proof that $l^1$ is isometrically isomorphic to $c_0^*$ correct?

This is a classic exercise of functional analysis, but I do not fully understand it after reading many answers in textbooks. So I am trying to reorganize the proof step by step in details. I am hoping ...
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Surface with prescribed first fundamental form

Consider a Riemannian manifold $(S,g)$ of dimension 2. What can we say about the possibility of an isometric immersion of this surface into $\mathbb{R}^3$? Of course this is not unique, even up to ...
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2answers
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$T$ is an isometry if and only if $\langle Tx, Ty \rangle = \langle x, y \rangle$

I want to prove: A linear mapping $T:X \to Y$ between two pre-Hilbert spaces is an isometry if and only if the inner products $\langle Tx, Ty \rangle = \langle x, y \rangle$ for all $x, y \in ...
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Is the nonnegative orthant isometric to itself under orthogonal mapping?

Problem description: (Informal). I simply want to know if there exists a necessary and (or at least) sufficient condition for an orthogonal matrix to map every point of the nonnegative orthant (say ...
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Is the normed linear space $X$ isometrically isomorphic to $Y$, if there is a linear operator $T: X \to Y$ such that $\|T\|=1$?

My question is if $\|T\|=1$ a sufficient condition for isometric isomorphism. If yes, how to prove it? If not, which additional conditions are needed? Perhaps $\|T^{-1}\|=1$ is necessary?
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1answer
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Example of not quasi-regular mapping

Example : (1) If $f : \mathbb{R}^2\rightarrow \mathbb{R}^2,\ f(x)=C\cdot x$ is dilation, then it is bi-Lipschitz map. (2) More generally, we consider an inversion $f:\mathbb{R}^2-\{o\}\rightarrow \...
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1answer
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Every finite dimensional normed (real) space is isometrically isomorphic to $\mathbf R^n$

Let $(E,\|•\|)$ be a n-dimensional real normed space. So there exists $T:E\longrightarrow\mathbf{R}^n$ linear isomorphism that for a well known result is also a topological homeomorphism. How could I ...
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1answer
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Prove that the following R linear map is an isometry.

Let $T \colon \mathbb{C} \to \mathbb{C}$ be an $\mathbb{R}$-linear map (where $\mathbb{C}$ is identified with $\mathbb{R}^2$ as usual). (a) Show that there exist complex numbers $\lambda, \mu \...
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1answer
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Are Riemannian manifolds only referred to as diffeomorphic if the diffeomorphism is an isometry?

Let $M$ and $N$ be Riemannian manifolds. My understanding is that strictly speaking, a diffeomorphism $\phi:M \to N$ only acts on the smooth manifold structure, not the metric tensor. But there is a ...
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Permutation matrix preseves volume of a set in Euclidean space

Suppose I have a set $U$ in $\mathbb{R}^n$. For example, $[3,5]$ in $\mathbb{R}$. We know that the volume (length) in this example is $2$. Suppose I have a permutation matrix $P$, I want to show ...
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1answer
59 views

isometry preserves the coefficient of second fundamental form?

isometry preserves the coefficient of 1st fundamental form. How about the coefficient of 2nd fundamental form? Is there any counterexample? Thanks for your answer in advance. Indeed, if F is isometry ...
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Let $Y$ and $Z$ be completions of $X$, then there exists an isometry from $Y$ onto $Z$.

Proof Attempt: Let $X$ be dense on metric spaces $Y$ and $Z$. Let $\overline{X}=Y$ and $\overline{X}=Z$. Suppose that $\{a_n\}$ is a sequence in $A$ such that $\{a_n\}\rightarrow y,z$ such that $y\in ...
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2answers
29 views

Distance and origin preserving map between two normed vector spaces

Let $V$ and $W$ be two normed vector spaces and let $f: V \rightarrow W$ be a map that preserves distance as well as the origin, that is : $ \forall x,y \in V \quad \|f(x)-f(y)\|_W=\|x-y\|_V \quad $ ...
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2answers
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Is a non-euclidean-norm preserving map necessarily linear?

Let $V$ and $W$ be two normed vector spaces and let $f:V \rightarrow W$ be a norm preserving map. I know that if both norms correspond to some inner product then $f$ is necessarily linear, but I can't ...