## Tagged Questions

0answers
45 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 …
3answers
229 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 …
0answers
86 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 isom …
1answer
94 views

### Properties of $S_2$ and the plane and $[-1,1]^2$ [closed]

Is the sphere $S_2$ isometric / isomorphic / diffeomorphic / homeomorphic to the plane? Is the sphere $S_2$ minus a point isometric / isomorphic / diffeomorphic / homeomorphic to …
1answer
358 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 i …
1answer
219 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 …
1answer
111 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 \$ …
2answers
315 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 \$\ …
0answers
427 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 dis …
1answer
499 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 …
2answers
375 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). …
1answer
435 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}) = \math …
0answers
308 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 surjectiv …
2answers
614 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 …
1answer
361 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, …

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