All Questions
Tagged with isometries reference-request
11 questions
4
votes
1
answer
380
views
Where to begin in Computational Group Theory?
I'm coding a small application that looks for periodic solutions to the gravitational n-body problem. I'm trying to better understanding the symmetries of solutions, which is made up of the product of ...
- 297
4
votes
1
answer
292
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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 ...
- 10.6k
5
votes
1
answer
293
views
All-set-homogeneous spaces
This is a follow-up to the question of Joseph O'Rourke Which metric spaces have this superposition property?
A metric space $X$ will be called all-set-homogeneous if for any subset $A\subset X$ any ...
- 45k
1
vote
1
answer
523
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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 $E_i$ ...
- 43
6
votes
2
answers
208
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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: $...
- 6,741
1
vote
2
answers
383
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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 $d>2$, ...
- 6,741
13
votes
3
answers
2k
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-...
- 133
6
votes
2
answers
483
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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 ...
- 2,371
16
votes
5
answers
904
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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) ...
- 151k
7
votes
1
answer
497
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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 ...
- 5,891
1
vote
2
answers
414
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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 \...
- 10.5k