20
votes
Accepted
Are all maps $\mathbb{R}^2 \to \mathbb{R}^2$ with fixed singular values affine?
Answer modified on 1 February 2020:
It's not true 'locally' in the sense that non-affine $f$'s satisfying this system of PDE can be constructed on some open sets in $\mathbb{R}^2$. This first order,...
- 108k
9
votes
Accepted
Alexandrov's generalization of Cauchy's rigidity theorem
The following is Theorem 27.2 of Igor Pak's book Lectures on Discrete and Polyhedral Geometry (which in general is a very nice resource for these sorts of questions):
Let $P,Q\subset\mathbb{R}^d$ (...
- 13.6k
9
votes
Are all maps $\mathbb{R}^2 \to \mathbb{R}^2$ with fixed singular values affine?
I would like to propose a simple local example:
Consider the map in polar coordinates, $\mathbb C\to \mathbb C$ that takes a complex number $z=e^{2\pi i \theta}r$ to $e^{(\sigma_1/\sigma_2)\cdot 2\...
- 28.9k
6
votes
Accepted
Cartesian monoidal star-autonomous categories
[I'm going to assume $S' \cong S$, which holds in every symmetric monoidal $*$-autonomous category. (See e.g. Lemma 5.6 of this paper.) This applies here since cartesianness implies symmetry. Part of ...
- 6,406
5
votes
Alexandrov's generalization of Cauchy's rigidity theorem
Wikipedia is correct. This is discussed in Alexandrov's book "Convex polyhedra" in Section 3.6.5.
- 6,337
5
votes
Alexandrov's generalization of Cauchy's rigidity theorem
This may help:
Bauer, C.
"Infinitesimal Rigidity of Convex Polytopes."
Discrete Comput Geom (1999) 22: 177. https://doi.org/10.1007/PL00009453
"Aleksandrov [1] proved that a simple convex $d$-...
- 151k
5
votes
Accepted
Cocycle superrigidity
Yes.
Instead of writing down explicitly such a cocycle (which is not hard), let me take this opportunity to explain how to think of such objects in a "cooridnate free" manner.
Given a cocycle $c:\...
- 11.6k
5
votes
Accepted
Constant Gaussian curvature disks
This is an addendum to the proofs by Anton and Deane, completing the missing part of the argument.
Lemma. Let $D$ be the closed unit disk and $f: D\to S^2$ an immersion such that $f(\partial D)$ is a ...
- 12.3k
3
votes
Accepted
Shrinking a disk with fixed differential
Here are a few comments that you might find useful, though they don't completely solve the problem. First, using symmetries of the problem, you can easily reduce to the case that $f$ is mapping the ...
- 108k
3
votes
Counterexample to mostow rigidity theorem
Yes, I believe there are many.
For example, if you think of hyperbolic $2$-space as a geodesic subspace of hyperbolic $3$-space, any group of hyperbolic isometries of hyperbolic $2$-space extends ...
- 44.4k
3
votes
How to correctly state Cauchy's rigidity theorem?
Yes, formally speaking the statement is incorrect.
The correct assumption is: there is a continuous bijection between the surfaces of two polyhedra that maps each facet to a facet by isometry.
...
- 45k
3
votes
Accepted
Forbidden minors of a graph with treewidth at most 4
I have a copy of Sander's PhD thesis. Counting $K_6$, there are actually $76$ excluded minors for treewidth at most $4$ (found by computer) in the thesis, but it is unknown if this list is complete (...
- 32.1k
3
votes
Forbidden minors of a graph with treewidth at most 4
This is not a complete solution in the sense of a complete list, just a description to get an easy example of a graph of treewidth $5$ that does not have $K_6$ as a minor.
As far as I know, the ...
- 105
1
vote
Constant Gaussian curvature disks
The answer is yes.
Since curvature is 1, there is an isometric immersion $\iota\colon D\looparrowright \mathbb{S}^2$. Note that the curve $\iota(\partial D)$ has constant curvature, therefore $\iota(\...
- 45k
1
vote
Show that duality functor is anti-monoidal
First of all, by Mac Lane Coherence Theorem we may assume that $\mathcal{C}$ is strict. Therefore we may omit associativity and unit constraints and we are left to check that
$$J_{U,W \otimes V} \circ ...
- 718
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