# Questions tagged [isometry]

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

584 questions
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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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### 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 $\dim(\operatorname{range}(T)) = \dim(\operatorname{range}(\sqrt{T^*T}))$

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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### 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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### 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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### 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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### 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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### 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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### 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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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 $\left|\left|x\right|\right| = \left|\... 1answer 30 views ### Iteration of a parabolic transformation Let$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^... 0answers 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? 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$...
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 ...
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 ...