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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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Combination of 1981 glide reflections in $\mathbb{E}^2$ still a glide reflection?

I was wondering if the combination of 1981 glide reflections over different lines is still a glide reflection over a line in $\mathbb{E}^2$ (so every glide reflection can be over a different line). Or ...
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If $||x_0-x_1||=||y_0-y_1||$, there exists an affine isometry such that: $f(x_0)=y_0 , f(x_1)=y_1$

We define: Affine transfomation: $f:\mathbb{R}^n\to\mathbb{R}^n$ as follows: $f(x)=Ax+b$ as $A\in \mathbb{R}_{nxn}$ and $b\in\mathbb{R}^n$. Isometry: $\forall x,y\in\mathbb{R}^n: ||f(x)-f(y)||=||x-...
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Triangles ABC and A'B'C 'are congruent. [on hold]

The data are points A (1, 0), B (2, 0), C (1, -1), A '(1, 1), B' (2, 1), C '(x, y). Determine x and y so that the triangles ABC and A'B'C 'are congruent and designate the isometry to perform one on ...
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Every isometry of a compact metric space into itself is onto

Despite of being aware of the duplicates of this question, I am posting it just to have my approach verified. Let $f:X \to X$ be an isometry on $X$. By definition, when $f(a)=f(b)$, then $ d_X(f(a),...
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isometry and orthogonality proof

If I have a relation (assuming $\vec{f}$ is one-to-one with $\det(\nabla \vec{f})>0$) appicable to all points from the domain of $\vec{f}$ which a regular region (a closed region with piecewise ...
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Examples of non-unitary isometries on finite dimensional Hilbert spaces?

I was reading the question A Finite Dimensional non-Unitary Isometry?, which gives an example of a non unitary isometry which is a map $T: R \rightarrow R^2 $. This question is based on a previous ...
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Prove that hyperbolic isometry with constant distance is the identity

Let $f$ be an isometry of the Poincaré half-plane model of two-dimensional hyperbolic geometry, denoted by $\mathbb{H}^2$. Prove that if the distance $d(z,f(z))=c$ for some constant $c\geq0$ for all $...
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A group of isometries

I've just started to learn isometries and groups, and I'm currently learning isometries of bounded (finite) figure. I'm confused that For a bounded figure $F \in \Bbb R^2$ and G is a set of all ...
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Making proof rigorous that isometries of $\mathbb{R}^n$ are bijective using balls

I was wondering how the proof of the fact that isometries of $\mathbb{R}^n$ are bijective could be made rigorous using the following argument. I understand that there is a more general (topological) ...
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Examples of isometry group

I would like to find some isometries groups G of a $\delta$-hyperbolic spaces such that $ \rho(G)=inf_{g \neq e}(||g||)>4\delta$. I already find $\mathbb{Z}$ acting on $\mathbb{R}$ that verify that....
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Isometric embedding of standard simplex

The standard $n$-simplex is the subset of $\mathbb R^{n+1}$ given by $\Delta^n = \left\{(t_0,\dots,t_n)\in\mathbb{R}^{n+1}~\big|~\sum_{i = 0}^n t_i = 1 \text{ and } t_i \ge 0 \text{ for all } i\...
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38 views

Construct a natural map $l_1 → l_{\infty}^∗$

I know from Show that $(l_1)^* \cong l_{\infty}$ that $l^∗_1$ is isometrically isomorphic to $l_{\infty}$. Indeed we can show that a map $L: l_{\infty} \rightarrow (l_1)^*$ given by $$L(x)(y) = \sum_{...
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Complex-analytic isometry of hyperbolic ball and half-space

I'm trying to prove that the $n$-dimensional Poincaré ball and half-space models of hyperbolic space are isometric. Here the Poincaré ball $\mathbb B^n_R$ ($n$-ball of radius $R$) has the Riemannian ...
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Showing segmented points of colinear lines are colinear

Let $ A_1, B_1, C_1 $ and $ A_2, B_2, C_2 $ be 2 pairs of collinear points such that $ \frac{A_1C_1}{C_1B_1} = \frac{A_2 C_2}{B_2 C_2} = k $ Let $ B $ and $ A $ be points of the segments $ B_1 B_2 $...
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Showing two spaces not isometrically isomorphic

Let $X$ be a real Banach space. Consider the direct sum $X\oplus X\oplus X$ with the norm $$\|(x,y,z)\|_1=\|x\|+\|y\|+\|z\|\text{ for all }x,y,z\in X.$$ I want to show that this space is not ...
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Mapping $\Bbb N\to\Bbb Q$

I want to set up a map from $\Bbb N\to\Bbb Q.$ Take $\Phi_S(x)=e^{(S/\ln(1-x))}$ and $M_T(1-x)=\Phi_S(x); S,T\in\Bbb N.$ Set $\Phi_S(x)=M_T(x)$ to obtain algebraic $x$ coordinates. If $x$ happens ...
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Is there a distance preserving map from the metric $d(a,b) = \log(|ab| + 1)$ to a Euclidean metric?

Let $M$ be a metric space, where the point are points in the euclidean plane and $$d_M(a, b) = \log(|ab|+1)$$ where $|ab|$ is the euclidean distance from $a$ to $b$. This is a metric since $x \mapsto \...
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Riesz representation theorem: show isometry

Let $X$ be a Hilbert space and $J:X\rightarrow X',J(X):=(\cdot,x)$ where $X'$ is the dual space of $X$. I have to show that $\|J(x)\|_{\sup}=\|x\|$. ''$\leq$'' is clear by the Cauchy-Schwarz ...
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1answer
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Prove that this group action is continuous from $S\times\Bbb{R}^2\to\Bbb{R}^2$

Let me give some description about the notations used- S denotes the collection of all transformations on $\Bbb{R}^2$ by integer coordinates. i.e. $S=\{t_v|t_v(x)=x+v,\forall x\in\Bbb{R}^2; \forall v\...
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Linear isometry and its trace

(We're in $\mathbb{R}^3$) What can we say about type of linear isometry $F : \mathbb{R}^3 \to \mathbb{R}^3$ if trace of $\mathrm{m} (F)$ is $-2$ or $\frac{1}{\sqrt{2}}$ or $\sqrt{2}$? Which one of ...
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Generalisation: isomorphism of sequences spaces

As is well known, the isometric isomorphism $(c_0)^* \cong \ell^1$ holds. Is there an analogous statement for general $L^p$ spaces? Perhaps a good start would be to wonder about generalizations of ...
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Is the adjoint of an isometry an isometry?

Consider a linear operator $V$ between two Hilbert spaces $H_A$ and $H_B$, we say that $V:H_A\rightarrow H_B$ is an isometry if $$ V^*V=\mathbb{1}_A$$ if $V$ is an isometry, does it always hold that ...
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1answer
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Is the Hodge dual the unique map which commutes with exterior powers of isometries?

Let $V$ be a real oriented $d$-dimensional inner product space, $d \ge 3$. For $1 \le k \le d-1$, the Hodge dual map $\star: \bigwedge^k V \to \bigwedge^{d-k} V$ commutes with orientation-preserving ...
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Orbit of a point and symmetries of a specific graph

Given is the graph: I am interested in determining the orbit of the point 1 and also to determine the amount of symmetries that fix each of the points 1, 2 and 3? My approach: Notice that the ...
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29 views

Projection from point onto plane

Let the plane is $\prod:\vec{x}\bullet\hat{n}=d$ be a plane, where $\vec{x}\in R^{3}$ and $d\in R$, then can I show using norm and Cauchy-Schwarz inequality that projection of point $x$ onto the ...
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1answer
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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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1answer
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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
129 views

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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0answers
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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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0answers
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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
31 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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0answers
42 views

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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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
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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
56 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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0answers
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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
24 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
52 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
19 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
45 views

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
46 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
30 views

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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19 views

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
197 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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0answers
24 views

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
130 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 ...