# 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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### 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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### 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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### 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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### 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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### 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,...
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 ...