The isometries tag has no wiki summary.

**8**

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**1**answer

115 views

### When is a continous $\epsilon$-isometry of the sphere surjective?

Equip $\mathbb S^n$ with the standard round metric. Let $f : \mathbb S^n \to \mathbb S^n$ be a continous map satisfying $\vert d(f(x),f(y)) - d(x,y)\vert \leq \epsilon$.
Is $f$ is surjective for all ...

**5**

votes

**1**answer

292 views

### The surjectivity of the exponential map for the isometry group

Little is known on general conditions guaranteeing that the exponential map between a Lie algebra and an associated Lie group is surjective.
Let $M$ be a noncompact connected Riemann manifold, and ...

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votes

**0**answers

19 views

### norm is invariant under scaling for subspace of wiener space

Consider the subspace H_1 of $C_0(0,\infty)$, where $\phi=\int_0^t\dot{\phi}(s)ds$ and $\int_0^{\infty}{\dot{\phi}}^2ds<\infty$. The transformation is $(T\phi)(t)=t\phi(\frac{1}{t})$. How to show ...

**1**

vote

**2**answers

365 views

### Isometric embeddings of metric spaces in Hilbert spaces

There are plenty of isometric embeddings of metric spaces in Banach spaces. Nevertheless, I have been unable to find any significant result on isometric embeddings into Hilbert spaces. My question is: ...

**6**

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**0**answers

192 views

### About Palais' remark that an isometry of Riemannian manifolds does not induce an isometry of the Hilbert manifolds of curves

In the paper ``Morse theory on Hilbert manifolds'' (1963), on page
326, Richard Palais makes a remark that if $\phi\colon V \to W$ is an
isometry (of submanifolds of $\mathbb{R}^n$), then this does ...

**4**

votes

**1**answer

310 views

### Isometry group of pseudo Riemannian manifold always a Lie group? (Myers-Steenrod)

Myers-Steenrod states that the isometry group of a Riemannian manifold is a Lie group. Is that also true for pseudo Riemannian manifolds? I didn't find anything related to that.
Cheers

**1**

vote

**1**answer

125 views

### Embedding of Two Objects Into Higher Dimensions With Their Sum

Given two vector sets, $\vec x_i$ and $\vec y_i$ (for $i$=1,2,...N, but the dimensionality of each vector can be more than N), let their sum set be $\vec z_i = \vec x_i + \vec y_i$. It's easy to ...

**14**

votes

**2**answers

378 views

### Repeated random two-steps in $\mathbb{R}^3$: unbounded?

I created a random isometry $T$ of $\mathbb{R}^3$ by generating
a random orthogonal matrix $M$,
uniformly distributed among all such,
and a random displacement $v$, whose coordinates
are drawn from a ...

**0**

votes

**0**answers

240 views

### Surface locally isometric to a sphere.

If for any two points p,q in a regular, compact surface $S\subseteq R^3$, there exists an isometry $f:S\rightarrow S$ s.t. f(p)=q. How to prove that $S$ is locally isometric to the sphere?

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**0**answers

96 views

### Isometric decomposition

Any progress on the following: Can a closed disc in the plane be partitioned into three disjoint sets which are pair-wise isometric, i.e. each set is an image of the others an isometry?

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**3**answers

265 views

### Which metric spaces have this superposition property?

Let $A \subset X$ and $B \subset X$ be two isometric subsets of a metric space $X$.
So there is an isometry $f: A \to B$.
Say that a metric space $X$ has the superposition property (my terminology)
...

**4**

votes

**1**answer

273 views

### Open problems about CMC hypersurfaces with symmetries?

Recently, Andrews and Li announced a complete classification of CMC ($H=const.$) tori in $S^3$, confirming a conjecture of Pinkall and Sterling. Their main result is that any such torus is ...

**2**

votes

**1**answer

135 views

### Partial isometries making families of linearly independent vectors orthogonal

Suppose I have a family of $n$ linearly-independent elements $v_i$ of the Hilbert space $\mathbb{C}^m$, which are not necessarily orthogonal. Can I always find a partial isometry $f: \mathbb{C} ^m \to ...

**11**

votes

**1**answer

428 views

### Possible isometries of a positively curved $S^2\times S^2$

Just to put things in perspective, recall that the Hopf Conjecture asks whether $S^2\times S^2$ admits a metric of positive sectional curvature. By the work of Hsiang-Kleiner, it is known that, if ...

**1**

vote

**2**answers

351 views

### All the isometries of $\mathbb{C}^n$ into itself are made like these

This is again a request for references. I'd appreciate a pointer to any published proof of the following:
Proposition. Given $n \in \mathbb{N}^+$, let
$\Phi$ be a function $\mathbb{C}^n
> \to ...

**2**

votes

**1**answer

604 views

### Isometry groups of Riemannian submersions with totally geodesic fibers

Suppose $F\to M\stackrel{\pi}{\to} B$ is a Riemannian submersion with totally geodesic fibers, all manifolds compact. In general, unless $M=B\times F$ is a Riemannian product, the isometry groups of ...

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**0**answers

339 views

### Surjectively isometric normed spaces: Hamel vs (extended) Schauder dimension

Bonjour/bonsoir à toutes et à tous.
This may really be a very basic question, but... Let $\mathbf{X} \equiv (X, \|\cdot\|_X)$ and $\mathbf{Y} \equiv (Y, \|\cdot\|_Y)$ be surjectively isometric (1) ...

**3**

votes

**2**answers

411 views

### “Measuring” how far is one Banach space from being surjectively isometric to another

Bonjour/bonsoir à toutes et à tous.
Assume that $\mathbf{V} \equiv (V, \|\cdot\|_V)$ and $\mathbf{W} \equiv (W, \|\cdot\|_W)$ are Banach spaces (over the real or complex field).
Question 1. What ...

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**0**answers

482 views

### Homometric $\Rightarrow$ isometric?

Suppose you know that there is a mapping between
two Riemmanian manifolds $M_1$ and $M_2$ such that,
for each $x_1 \in M_1$, the (codimension-1) measure of the set of points
at distance $d$ from $x_1$ ...

**2**

votes

**1**answer

619 views

### When do 0-preserving isometries have to be linear?

Let $\langle \mathbf{V},+,\cdot,||.|| \rangle$ be a normed vector space over $\mathbb{R}$.
Let $f : \mathbf{V} \to \mathbf{V}$ be an isometry that satisfies $f(\mathbf{0}) = \mathbf{0}$ .
What ...

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votes

**1**answer

441 views

### Must a surjective isometry on a dual space have a pre-adjoint?

Background: Let $X$ be a Banach space. We know a linear map $h$ is a surjective isometry of $X$ if and only if its adjoint $h^*$ is a surjective isometry of $X^*$.
In general, a linear map $g:X^* ...

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869 views

### Terminology: “cocompact”

Let $M$ be a Riemannian manifold such that its isometry group $G=\textrm{Iso}(M)$ is a Lie group, and let $\Gamma$ be a subgroup of $G$.
1) What does the phrase "$\Gamma$ is a cocompact group of ...