# Questions tagged [isometries]

The isometries tag has no usage guidance.

74
questions

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### Uniqueness of function with range $\mathbb{S}^2$ under a constraint

Assume $g,f\colon A\subset\mathbb{R}^M\rightarrow\mathbb{S}^2$ are two bijective functions defined on the set $A$. Now assume a constraint $C$: $\forall x,y\in A, \exists R\in SO(3)\colon Rf(x)=f(y)\...

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76 views

### Isometric stratification preserves volume?

Let $K\subset \mathbb{R}^k$ be a non-empty compact subset let $f:K \to K$ be Lipschitz and surjective. If, moreover, $f$ is an isometry then clearly $f$ preserves the Lebesgue measure of $K$.
I ...

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27 views

### proof of the following theorm on simply-connected, complete indefinite Kahler manifold

can anyone help me prove the following therorm
If $c\in \mathrm{I\!R}$ every connected, simply-connected, complete
indefinite Kahler manifold of complex dimension $n$, of index 2s and of
constant ...

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**1**answer

183 views

### Separable Banach spaces isometric to quotient of a Banach space

We know that every separable Banach space is isometrically isomorphic to a quotient space of $(\ell^1,\|.\|_1)$. We also know that the norm defined by $\|x\|=(\|x\|_1^2+\|x\|_2^2)^{1/2}$ for all $x\in ...

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71 views

### 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 ...

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82 views

### 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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112 views

### 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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508 views

### 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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95 views

### 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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62 views

### 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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152 views

### 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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95 views

### 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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**1**answer

525 views

### 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 ...

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56 views

### 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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119 views

### 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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130 views

### 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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575 views

### If $E\oplus_\phi E \cong E\oplus_\psi E,$ does it imply that $\phi= \psi$?

Let $E\neq \{0\}$ be a Banach space.
For each $p\in[1,\infty), $ we define
$$E\oplus_p E = \{(x,y): x\in E, y\in E, \|(x,y)\| = \sqrt[p]{\|x\|^p + \|y\|^p}\}.$$
Let $F$ be another Banach space.
By $E\...

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36 views

### On the minimum distance along an orbit

Let $\Gamma$ be a nontrivial group of isometries of $\mathbb{S}^n$, $n \geq 2$, acting properly discontinuously. For $p \in \mathbb{S}^n$, define
$$r(p) = \min_{g \in \Gamma \setminus\{e\} } d(p, g(p)...

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501 views

### Laplace-Beltrami and the isometry group

H$\vphantom{a}$i. Consider the Laplacian on $\mathbb R^n$,
$$
\Delta=\partial_i^2
$$
It is easy to prove that the most general differential operator that commutes with rotations and translations is ...

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140 views

### Are all symmetries of the Dirichlet functional isometries?

This is a cross-post from MSE (no answer there).
Let $M,N$ be oriented $d$-dimensional Riemannian manifolds, $M$ compact*, and let $f:M \to N$ be smooth.
Consider the Dirichlet energy functional: $...

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185 views

### 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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418 views

### 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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309 views

### What is general expression for the moment map of a Kaehler Hamiltonian G-manifold

A Kaehler Hamiltonian G-manifold is a Kaehler manifold with a Hamiltonian G-action, i.e., a G-action generated by a moment map. In particular, the Killing vector fields which generate the G-action are ...

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312 views

### 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 $\{-1,1,0\}$, ...

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641 views

### Tweetable way to see Riemannian isometries are harmonic?

$\newcommand{\al}{\alpha}$
$\newcommand{\euc}{\mathcal{e}}$
$\newcommand{\Cof}{\operatorname{Cof}}$
$\newcommand{\Det}{\operatorname{Det}}$
Smooth Riemannian isometries are harmonic. Can one conclude ...

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751 views

### A differentiable isometry is smooth?

I posted this question in MSE but got no response (even after giving a bounty), so I am trying here.
Let $M,N$ be smooth $d$-dimensional Riemannian manifolds.
Suppose $f:M \to N$ is a differentiable ...

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165 views

### Homogeneous subsets of the sphere

Let $S$ be a (unit) sphere in a Hilbert Space $H$ with $\dim H \ge 3$. Let $A \subset S$ have the following properties:
$A$ is connected;
The affine hull of $A$ is the whole space;
For every $x,y\in ...

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407 views

### There is no arcwise isometry from a high dimensional manifold into a low dimensional manifold

$\newcommand{\al}{\alpha}$
$\newcommand{\ga}{\gamma}$
$\newcommand{\e}{\epsilon}$
Let $X,Y$ be Riemannian manifolds, such that $\dim(X) > \dim(Y)$.
I am trying to prove the following statement (...

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287 views

### Are metric isometries smooth at the boundary?

Let $M,N$ be smooth Riemannian manifolds with boundary (In particular, we assume the boundaries are smooth).
Suppose we have a map $\phi:M \to N$ which satisfies the following properties:
$$(1) \, \,...

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293 views

### Does nonexpanding map between manifolds decrease volume?

(This question is a special case of a question I asked at SE, which got no answer there)
Let $M,N$ be diffeomorphic connected compact Riemannian manifolds, and let $f:M \to N$ be a surjective ...

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307 views

### Bending Beltrami Pseudosphere

The Beltrami Pseudosphere
$$[x = a \sin p \cos t , y= - a ( \cos p + \log \tan p/2 ) , z= b+ a \sin p \sin t \; ], (.1 <p<\pi/2), (0< t< 2 \pi), \; (b>a) $$
is bent to a non-...

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780 views

### Are there some intrinsic invariants of surfaces other than Gaussian curvature?

The principal curvatures of a surface is denoted by $\kappa_{1}, \kappa_{2}$.
Let $P(x,y)$ be a polynomial with real coefficients. Assume that $P(\kappa_{1}, \kappa_{2})$ is an intrinsically ...

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148 views

### How isometric action on Riemannian manifold acts on cut locus

Assume that $M$ is a simply connected closed Riemannian manifold with no boundary and nonnegative sectional curvaure Assume that ${\bf Z}_n=(g),\ n\geq 3$ acts on $M$ isometrically. Then if $gx=x$, i....

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183 views

### isometric action on the $n$-sphere

Let $S^n$ be the $n$-sphere. If $n=2k+1$ is odd, then we can identify $S^n$ as a subset of $\mathbb{C}^{k+1}$. We define the $S^1$ action on $S^n$ by multiplication, namely
$$ \Psi \colon S^1 \times ...

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79 views

### Finding the infimum using a piecewise isometry

Given a finite set of unit circles in the plane such that the area of their union $U$ is $S$, what is the largest possible bound $kS$ for some constant $k$ such that there exists a subset of mutually ...

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841 views

### Isometry group of a compact hyperbolic surface

Consider a compact surface $M$ of genus $g \geq 2$ with a metric of constant negative curvature. My question is, is it known under what sorts of sufficient conditions such a metric will have non-...

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513 views

### Are the Sasaki metrics on tangent and cotangent bundle isomorphic?

Let $(M,g)$ be a Riemannian manifold. Then there is the well-known
Sasaki metric that makes $(TM,\hat{g})$ a Riemannian manifold. In a
similar way, one can construct a Sasaki metric $\bar{g}$ on the
...

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97 views

### Commutator subgroup of rotational symmetries of the hypercube

I would like to know which is the commutator subgroup of the group of rotational isometries of the $n$-dimensional cube. The group i am talking about is the subgroup of $\{ \pm 1 \} \wr S_n$ ...

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333 views

### Products of elliptic isometries

A well-known property on groups acting on trees is:
Theorem: Let $T$ be a tree and $g,h \in \mathrm{Isom}(T)$ two elliptic isometries. If $\mathrm{Fix}(g) \cap \mathrm{Fix}(h) = \emptyset$ then the ...

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673 views

### Under what conditions a linear automorphism is an isometry of some norm?

Assume $V$ is a finite-dimensional vector space over $\mathbb{R}$, and $T: V \to V$ is a (linear) isomorphism.
When is it possible to construct a norm on $V$
making $T$ an isometry?
(Hopefully,...

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**3**answers

2k views

### Is there a global obstruction for a diffeomorphism to be an isometry?

Let $V$ be a finite dimensional vector space.
Let us call an automorphism $T:V\rightarrow V$ admissible if there exists an inner product $\langle , \rangle$ on $V$ making $T$ an isometry.
We know $T$...

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**1**answer

145 views

### Discrete subgroup of centralizer of transvections in isometries acts properly discontinuously

My question will rely on a clarification of a proof, which I simply don't understand.
Let us denote by $X$ a pseudo-riemannian symmetric space and define
$$
Z_{\mathrm{Iso}\left(X\right)}G(X) = \{\, ...

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636 views

### Isometries of some simple Cayley graphs

Consider a Cayley graph of a group $G$ with respect to a symmetric finite generating set $S$.
There are some obvious candidates to isometries of this graph - for example, translation by elements of $G$...

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182 views

### Obstruction to the existence of global isometries on a constant-curvature Riemannian manifold

Let $M$ be an $m$-dimensional simply connected Riemannian manifold that is not geodesically complete. Suppose $M$ has constant sectional curvature.
Because the curvature is constant, locally $M$ ...

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1k views

### Under what conditions $\|x-y\|=n\iff\|f(x)-f(y)\|=n.$ for $n\in\mathbf{N}$ implies isometry?

Let $X, Y$ be normed space and $f:X\to Y$ be a mapping. Assume that for all $n\in\mathbf{N}$, $$\|x-y\|=n\iff\|f(x)-f(y)\|=n.$$
Under what conditions this map will be an isometry?
Thanks

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431 views

### Isometries of Compact Semisimple Lie Groups

In this delightful question, the poster mentioned that the isometry group of a compact Lie group $G$, equipped with the metric from the Killing form, is $G\times G/Z(G)$, where $Z(G)$ is the center of ...

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142 views

### Symmetries higher dimensional cube fixing subcubes

Which is the group of rotational isometries of an $n$ dimensional hypercube fixing an $m$ dimensional element (an $m$ dimensional subcube)? I know for example that it is $A_n$ for $m=0$ (symmetries ...

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610 views

### Gromov-Hausdorff convergence for non-compact metric spaces

Let $(X_i,p_i)$, $(X,p)$ be pointed connected proper metric spaces (i.e. the closures of balls are compact). Are the following two statements equivalent?
$\forall r > 0: \bar{B}_r(p_i) \stackrel{...

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504 views

### If all balls around two points are isometric… — manifold version

This question is a natural follow-up of this other question, asked earlier today by wspin.
Let's say that a metric space $(X,d)$ has two poles if:
there are two distinct points $x$, $y$ such that ...

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885 views

### If all balls at $x$ and $y$ are isometric is there an isometry sending $x$ to $y$?

Let $(X,d)$ be a metric space and $x,y \in X$. Assume that for all $r > 0$ the balls $B_r(x)$ and $B_r(y)$ are isometric.
Is it true that there exists an isometry of $X$ sending $x$ to $y$?