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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$C(K_1)\cong C(K_2)$ if and only if $K_1\cong K_2$

I had read the following statement in a book without proof. One of the directions is trivial, however the other is not: Let $C(K_j)$ the Banach space of continuous functions $f:K_j\to \mathbb{R}$ ...
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Can a finite Wasserstein metric on Euclidean support be embedded in a Euclidean space?

Thanks for everyone's help with understanding finite metric embeddings in Euclidean space. I have a follow-up question. Say we have the Wasserstein distance between $n$ distributions in Euclidean ...
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Spectral theorem for isometries

I will soon have an exam, and there is something that I simply don't understand: the spectral theorem for orthogonal matrices/endomorphisms (isometries). $\phi$ is an isometry $\Leftrightarrow$ There ...
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Is every isometry invariant measure on Euclidean space equivalent to Hausdorff measure?

One of the properties of Hausdorff measure on $\mathbb{R}^d$ is invariance under isometry. I wonder that if every isometry-invariant measure is similar to Hausdorff measure. To be rigorous, let $\mu$ ...
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Tangent developable is locally isometric to an open set in ℝ²

I'm trying to solve this problem about tangent developable from a differential geometry exam $\sf 1996\ Q3$: My working: First part $\ldots\ldots$deduce that $Σ$ is ruled (that is, each point of $Σ$ ...
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$V$ Euclidean iff there exist linear isometry that $f(v)=w$ for $|v|=|w|$

From the book 'a course in metric geometry' exercise 1.2.24 : Let $V$ be a finite-dimensional normed space. prove that V is Euclidean iff for any tow vectors $v,w\in V$ such that $|v|=|w|$ there ...
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Are there self-adjoint operators with eigenvalues 1, -1 that are not isometries?

Let $f$ be a self-adjoint operator in an euclidean metric vector space $(V, g)$. a) Prove that the following are equivalent. i) The only eigenvalues of $f$ are 0, 1. ii) $f$ is an orthogonal ...
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isometries as products of reflections [closed]

I am taking a course on geometric transformations in the 2-dimensional Euclidean plane. I have been told that all isometries (length-preserving transformations) are products of reflections about ...
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Extending linear isometry from subspace to whole space

Problem: Let $(V, \langle \cdot, \cdot \rangle_V)$ and $(W, \langle \cdot, \cdot \rangle_W)$ be finite-dimensional inner-product spaces, such that $\dim V\leq \dim W$, and let $U\subset V$ be a ...
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Veryshynin HDP Exercise 4.1.4: Equivalent definitions of isometry

I'm trying to show the following: Let $A$ be an $m \times n$ matrix with $m\geq n$. Prove that the following statements are equivalent. $A^{\top} A = I_n$. $P = A A^{\top}$ is an orthogonal ...
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Does a map that preserves distance imply isometry?

It feels intuitive for this to be true. Let us say I have a chart from $\mathbb{R^2}$ to some section of a manifold in $\mathbb{R^3}$. If the distance in $\mathbb{R^2}$ between two points is equal to ...
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Isometry between cone and cylinder [closed]

In a certain exercise I have been asked to find an isometry between a portion of the cylinder $S = \{ x^2+y^2 = 2: 0 < z < 1\}$ and the complete cone $S_* = \{x^2+y^2 = 2z^2: 0 < z < 1 \}$...
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Isometry between triangle and cone

In a certain exercise I have been asked to say if there is an isometry between the triangle $T =\{ z = 0, 0 < x, y, x + y < 1 \}$ and the cone $C = \{x^2 + y^2 =\frac{z^2}{4}, 0 < z < 2\}$....
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Prove $\ell^1 \cong c_0*$

To prove: $T:l^1 \to c_0^*, \langle Tx, y\rangle = \sum x_ky_k$ is an isometric isomorphism Is this proof correct? Particularly, I dont know if I proved injectivity correctly Proof: (i) linearity: let ...
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The isometry group of a metric space with the topology of pointwise convergence is a topological group

Let $(X,d)$ be a metric space and let $\mathrm{Iso}(X)$ denote the group of isometries of $(X,d)$, where the group operation is the composition. Equip $\mathrm{Iso}(X)$ with the topology of pointwise ...
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Showing an abstract surface (with a specific metric) is isometric to a helicoid

$\DeclareMathOperator{\sech}{sech}$I've been trying to understand a question in Barrett O'Neill's Elementary Differential Geometry that goes as follows: "Show that the geometric surface in ...
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Prove that a change-of-basis map is an isometry between complex hilbert spaces, to prove uniqueness of purifications

I am trying to prove the following For this I should use the following fact: Let $ρ_A = \sum_{i=1}^r p_i|e_i⟩⟨e_i|$, where $p_i$ are the nonzero eigenvalues of $ρ_A$ and $|e_i⟩$ corresponding ...
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Is a map that preserves torsion and curvature of all smooth curves an isometry?

In my introductory differential geometry class, we learnt that isometries preserve the torsion $\tau$ and curvature $\kappa$ of curves. I was wondering if the converse of this statement is true: if a ...
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Which groups are achievable as achievable as the scale factors of a manifold?

Let $(M, g)$ be a Riemannian manifold. I say $\lambda\in\mathbb R^\times$ is a scale factor of $M$ if $(M, g)$ is isometric to $(M, \lambda g)$. Composing isometries multiplies the scale factor, so ...
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Transform $3D$ grid to dimetric projection [closed]

I'm rendering rhombus textures in a $3D$ world: Here are the rhombus textures rendered in a grid in perspective projection with $FOV = 45^\circ$: Here is the same grid scene, but in dimetric ...
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Isometries of a submanifold with induced metric

I need help to find isometries of a submanifold of a semi-Riemannian manifold. To be crystal clear, let me start with what I mean by an isometry. $\textbf{Definition:}$ Let $(M,g)$ be a semi-...
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Showing that this embedding is isometric

Let $X$ be a metric space and $B(X)$ be the space of bounded functions $f: X \to \mathbb{C}$. Consider the embedding $J : X \to B(X)$ defined via $y \mapsto J(x)(y) = d(x, y) − d(x_0, y)$ for some ...
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