# Questions tagged [isometries]

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### Are two metric spaces isometric if they have the same $\varepsilon$-covering numbers for all $\varepsilon>0$?

Let $(E, d)$ be a metric space. For $\varepsilon>0$, we define two notions of $\varepsilon$-covering number as follows, i.e., $N_\varepsilon^o (E)$ is the smallest number of open balls whose radii ...
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### Is every 1-Lipschitz homeomorphism $f:X\to X$ from a compact metric space to itself an isometry?

I found a statement involving a homeomorphism $f:X\to X$ of a compact metric space $X$, with Lipshitz coefficient 1, i.e., a non-expansive map, and cannot think of an example where $f$ is not an ...
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### What is the minimum $n$ for which $\Bbb H^3$ can be isometrically embedded in $\Bbb R^n$ as a bounded set?

Consider the hyperbolic $3$-space $\Bbb H^3$ (i.e., the unique, simply-connected, $3$-dimensional complete Riemannian manifold with a constant negative sectional curvature equal to $-1$). The Nash ...
1 vote
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### Isometries of Hilbert space

It is easy to see that for any $x$ and $y$ on the unit sphere of a Hilbert space $H$ there exists a surjective isometry $U$ such that $Ux=y$. Does something more general also hold? That is, given two ...
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### Isometric embeddings of $c_0$ into metric spaces

Are there any nice and useful criteria or theorems which assert when a given metric space $M$ contains an isometric (not necessarily linear) copy of the Banach space $c_0$ or its unit ball $B_{c_0}$? (...
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### Are polyhedra with equilateral triangular faces rigid?

Convex polyhedra are rigid by Cauchy’s theorem. Steffen’s polyhedron is an example of a non-convex polyhedron that is flexible (i.e., non-rigid). However, it appears to have edges of different lengths....
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### Isometry between Minkowski space and Tangent space in an article by Stefan Waldmann [closed]

In the notes Geometric Wave Equations by Stefan Waldmann at page 70 they have Having a fixed Lorentz metric $g$ on a spacetime manifold $M$ we can now transfer the notions of special relativity, ...
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### Fixed points on spherical buildings

A crucial aspect of the Bruhat–Tits theory of affine buildings is the Bruhat–Tits fixed-point theorem, which, in one of many formulations, states that, if $\Gamma$ is a group of isometries of an ...
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### Palais's and Kobayashi's theorems on automorphism groups of geometric structures

My question concerns two results in the neighborhood of the standard theorem of Myers-Steenrod that isometry groups of Riemannian manifolds are Lie groups. Both appear in the first chapter of ...
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### Does a lifted functor on $\mathbf{1Met}$ preserve isometries?

Let $\mathbf{1Met}$ denote the category of metric spaces with distance bounded by $1$ and nonexpansive maps ($1$-Lipschitz functions). I call isometry a distance-preserving map (some people require it ...
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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 ...
1 vote
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### Stochastic Integral + conditional expectation

Let $\overline{\widehat{Z}_i} = \frac{E_i\left[ \int_{t_i}^{t_{i+1}}\widehat{Z}_sds\right] }{\Delta t_i}$ with $\widehat{Z}$ a square integrable process, $\Delta t_i := t_{i+1} - t_i$, and $E_i$ ...
1 vote
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### Ideal Ford domain (for finite index subgroup)

Let $G$ be a lattice Fuchsian group with parabolic elements, seen as a discrete subgroup of matrices $g= \begin{pmatrix} \alpha & \overline{\beta} \\ \beta & \overline{\alpha} \end{pmatrix}$...
1 vote
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### Characterization of extrinsic distance prevserving embedding (see the definition given!) from low dimensional Euclidean spaces to high dimensions

P.S. I asked the question on MSE more than a week ago, but didn't get any desired answer, so asking here. Let $m < n \in \mathbb{N}$. Let us equip $\mathbb{R}^m, \mathbb{R}^n$ with their ...
1 vote
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### How does this orthogonality follow from the map being an isometry?

This is a step of a proof in the book Variational Problems in Geometry by Seiki Nishikawa. I will ignore the background and change some of the statements and notations for simplicity. Let $(M,g)$ ...
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### Local isometry implies covering map: nonempty boundary case [closed]

The following theorem is well known in the literature: Let $M$ and $N$ be riemannian manifolds and let $f : M \to N$ be a local isometry. If $M$ is complete and $N$ is connected, then $f$ is a ...
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### What are the applications of the Mazur-Ulam Theorem?

Every bijective isometry between normed spaces is affine. This well-known and beautiful statement, the Mazur-Ulam Theorem, was proved in 1932, but the proof has been simplified and polished in years, ...
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### Spacetime symmetries

We know some nice space-time have a lot of symmetries. It is said that Minkowski spacetime has $$ISO(d-1,1)/SO(d-1,1),$$ de Sitter spacetime has $$SO(d,1)/SO(d-1,1)$$ and anti-de Sitter spacetime ...
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### Proximal isometries in CAT($-1$) metric space

Let $X$ be a rank $1$ symmetric space of non-compact type and $G$ its isometry group. $G$ is a semisimple linear algebraic Lie group of non-compact type with trivial center. Let $\rho$ be a ...
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### Pogorelov's rigidity theorem vs Cohn-Vossen rigidity theorem

There is the following rigidity theorem of Cohn-Vossen as stated on p. 86 of these lecture notes: http://www.math.brown.edu/~deigen/chern.pdf Any isometry between two closed smooth convex surfaces (...
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### Cohn-Vossen rigidity theorem in hyperbolic space

There is the following rigidity theorem of Cohn-Vossen as stated on p. 86 of these lecture notes: http://www.math.brown.edu/~deigen/chern.pdf Any isometry between two closed smooth convex surfaces in ...
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### Uniform distribution of points on Riemannian manifolds

Recently, I came across a beautiful paper by Arnol'd and Krylov (Uniform distribution of points on a sphere...) that contains the following theorem: Theorem: Let A and B be two rotations of the ...
1 vote
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### Mapping to distorted constant Gauss curvature surfaces of revolution

There are three questions here. We imagine a flexible membrane that is scrolled out so as to straighten it. 1) How can we find a surface isometrically mapped from a surface of constant negative Gauss ...
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### Minkowski isometries

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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### The isometry groups of flag manifolds

For any sequence of integers $0<n_1<...<n_k$, there is a flag manifold of type $(n_1, ..., n_k)$, which is the collection of ordered sets of vector subspaces of $R^{(n_k)}$ $(V_1, ..., V_k)$ ...
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1 vote
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### Conformal harmonic maps in high dimensions are scaled isometries

This is a cross-post from MSE (where I got no answer). It is well-known that conformal maps between $2$-dimensional Riemannian manifolds are harmonic. I discovered lately that in dimension $d>2$, ...
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### Do curvature differences obstruct a.e orientation-preserving isometries?

Is there an example of a pair $M,N$ of connected, oriented equidimensional Riemannian manifolds with the following properties: $M$ is everywhere non-flat, $N$ is flat. There exist a map $f:M \to N$ ...
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### Norms on $\mathbb{R}^d$ whose linear isometries are the hypercube group
It is a known fact that for any $2\neq p\in[1,\infty]$, the linear isometries for the corresponding norm $\|\cdot\|_p$ on $\mathbb{R}^d$ is the set of all square-matrices with entries in $\{-1,1,0\}$, ...