Questions tagged [isometry]

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

1008 questions
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Let, $A\subset\mathbb{R}^2$. Show that $A$ can contain at most one point $p$ such that $A$ is isometric to $A \setminus \{p\}$.

A challenge problem from Sally's Fundamentals of Mathematical Analysis. Problem reads: Suppose $A$ is a subset of $\mathbb{R}^2$. Show that $A$ can contain at most one point $p$ such that $A$ is isometric to $A \setminus \{p\}$ with the usual…
27
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Difference between orthogonal and orthonormal matrices

Let $Q$ be an $N \times N$ unitary matrix (its columns are orthonormal). Since $Q$ is unitary, it would preserve the norm of any vector $X$, i.e., $\|QX\|^2 = \|X\|^2$. My confusion comes when the columns of $Q$ are orthogonal, but not orthonormal,…
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How many ways are there to fill a 3 × 3 grid with 0s and 1s?

Extra conditions that put a formal solution out of my reach: the centre cell must contain a $0$, and two grids are equal if they have a symmetry, e.g. $$\left( \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 1\end{array} \right) = \left(…
Sup Bro
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If every five point subset of a metric space can be isometrically embedded in the plane then is it possible for the metric space also?

Let $X$ be a metric space with at least $5$ points such that any five point subset of $X$ can be isometrically embedded in $\mathbb R^2$ , then is it true that $X$ can also be isometrically embedded in $\mathbb R^2$ ?
user228169
19
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If two Riemannian manifolds can be isometrically immersed in each other, are they isometric?

Let $M,N$ be smooth compact oriented Riemannian manifolds with boundary. Suppose that both $M,N$ can be isometrically immersed in each other. Must $M,N$ be isometric? Does anything change if we also assume…
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Isometry group of a norm is always contained in some Isometry group of an inner product?

$\newcommand{\<}{\langle} \newcommand{\>}{\rangle} $Let $||\cdot||$ be a norm on a finite dimensional real vector space $V$. Does there always exist some inner product $\<,\>$ on $V$ such that $\text{ISO}(|| \cdot ||)\subseteq \text{ISO}(\<,\>)$ ?…
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Fixed Points Set of an Isometry

I'm reading Kobayashi's "Transformation Groups In Riemannian Geometry". I'm trying to understand the proof of the following theorem: Theorem. Let $M$ be a Riemannian manifold and $K$ any set of isometries of $M$. Let $F$ be the set of points of $M$…
Sak
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Non-trivial isometries of the left invariant metric on $GL_n$

Let $GL_n^+$ be the group of $n \times n$ real invertible matrices with positive determinant. Let $g$ be the left-invariant Riemannian metric on $GL_n^+$ obtained by left translating the standard Euclidean inner product (Frobenius) on $T_IGL_n^+…
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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 nonseparable spaces? That is, is there a known example…
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Is every geodesic-preserving diffeomorphism an isometry?

Let $M$ be a closed $n$-dimensional Riemannian manifold. Let $f:M \to M$ be a diffeomorphism and suppose that for every (parametrized) geodesic $\gamma$, $f \circ \gamma$ is also a (parametrized) geodesic. Must $f$ be an isometry? An equivalent…
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If there is an into isometry from $(\mathbb{R}^m,\|\cdot\|_p)$ to $(\mathbb{R}^n, \|\cdot\|_q)$ where $m\leq n$, then $p=q$?

Let $p,q\in [1,\infty)$. Note that $p,q\neq\infty$. Let $m\geq 2$ be a natural number. The paper Isometries of Finite-Dimensional Normed Spaces by Felix and Jesus asserts that if $(\mathbb{R}^m,\|\cdot\|_p)$ is isometric to $(\mathbb{R}^m,…
Idonknow
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How to define a Riemannian metric in the projective space such that the quotient projection is a local isometry?

Let $A: \mathbb{S}^n \rightarrow \mathbb{S}^n$ be the antipode map ($A(p)=-p$) it is easy to see that $A$ is a isometry, how to use this fact to induce a riemannian metric in the projective space such that the quotient projection $ \pi:…
Jr.
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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 $T$ is linear? If $T$ is surjective one…
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Any finite metric space can be isometrically embedded in $(\mathbb R^n,||\cdot||_\infty)$ for some $n$?

Let $X$ be a finite metric space, then is it true that $\exists n \in \mathbb N$ such that there exists an isometry from $X$ into $\mathbb R^n$, where $\mathbb R^n$ is equipped with the supremum metric?
user228168
9
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Some version of Itô isometry with conditional expectations

Let $B = (B_t)_{t \geq 0}$ be a Brownian motion, $ \mathcal{F}= (\mathcal{F}_t)_{t \geq 0}$ the natural filtration associated to $B$, $u \in L^2_{a,T}$ (that is, $u$ is an stochastic process $u = (u_t)_{0 \leq t \leq T}$ adapted to $\mathcal{F}$ so…
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