Questions tagged [rigidity]
The rigidity tag has no usage guidance.
20
questions
5
votes
2answers
396 views
Are all maps $\mathbb{R}^2 \to \mathbb{R}^2$ with fixed singular values affine?
Let $f:\mathbb{R}^2 \to \mathbb{R}^2$ be a smooth map whose differential has fixed distinct singular values $0<\sigma_1<\sigma_2$ and an everywhere positive determinant (which is the product $\...
5
votes
3answers
475 views
Alexandrov's generalization of Cauchy's rigidity theorem
Wikipedia states that A. D. Alexandrov generalized Cauchy's rigidity theorem for polyhedra to higher dimensions.
The relevant statement in the article is not linked to any source. The sources at the ...
3
votes
1answer
168 views
Tannaka-Krein reconstruction and rigidity
Let $\mathcal{C}$ be a rigid monoidal category together with a quasi-monoidal functor $\omega:\mathcal{C}\to\mathsf{vec}_{\Bbbk}$ to finite-dimensional vector spaces over a field $\Bbbk$, i.e. we have ...
5
votes
1answer
224 views
Cocycle superrigidity
Let $\Gamma$ be a group with a probability measure preserving action on $(X,\mu)$, and $H$ another group. Recall that a cocycle is a map $c:\Gamma\times X\to H$ such that $c(gg',x)=c(g,g'x)c(g',x)$. ...
4
votes
0answers
72 views
Conformal $L^p$ rigidity of Riemannian manifolds
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9
votes
1answer
729 views
A question about Mirzakhani et. al.'s algebraicity theorem
While the geodesic flow on a complete hyperbolic surface is ergodic, the closure of an individual orbit (a geodesic line) can take a complicated fractal-like shape. Nonetheless, there is an ...
24
votes
3answers
1k views
Which mapping class group representations come from algebraic geometry?
Let $\Gamma_g$ be the mapping class group of a closed oriented surface $\Sigma$ of genus $g$. There is a natural surjection $t \colon \Gamma_g \to \mathrm{Sp}(2g,\mathbf Z)$ which sends a mapping ...
2
votes
1answer
144 views
Is a rigid cycle a chordal graph?
There are two relevant questions:
(1) We know an edge set $C$ is a rigid cycle in $\mathcal{G}_2(n)$ if and only if $|E(C)|=2|V(C)|−2$ and $|F|≤2|V(F)|−3$ for every proper subset $F$ of $E(C)$. Thus, ...
4
votes
2answers
143 views
Which criteria guarantee an orthogonal circuit in $\mathbb R^3$ to be rigid?
For $n\ge4$, define an orthogonal circuit or O-circuit as a closed circuit of $n$ unit segments in $\mathbb R^3$ such that any two neighboring segments form a right angle. (Physically this could be ...
11
votes
1answer
461 views
quantitative version of the rigidity of the 2-sphere
I am looking for a quantitaive version of the following theorem:
A compact surface with $K\equiv 1$ is isometric to the round sphere.
Of course I get the Berger, Brendle-Schoen Theorem which insures ...
4
votes
1answer
162 views
Characterizing the rigidity of morphisms of smooth varieties
Let $X$ and $Y$ be smooth algebraic varieties over a field $k$ of characteristic $0$. For varieties we know that $X/k$ is rigid if and only if $H^{1}(X,T_{X})=0$. But $H^{1}(X,T_{X})$ also ...
10
votes
1answer
436 views
Topological rigidity for negatively curved manifolds?
I was wondering if two compact oriented manifold carrying a Riemannian metric with negative sectional curvature, whose fundamental groups are isomorphic, are necessarily diffeomorphic (or homeomorphic)...
3
votes
0answers
141 views
Rigidity vs Super-rigidity of representations (of Kähler/surface groups)
In the literature there are several definitions of "rigidity" (or "super-rigidity") of representations, adapted to the circumastances. I wonder what are the relations between them; I excuse in advance ...
11
votes
0answers
628 views
Definition of a uniformly bounded dual of a group
The unitary dual of a group $G$ is the set of equivalence classes of irreducible unitary representations of $G$ with the Fell topology. (This topology is defined using convergence of positive definite ...
1
vote
1answer
184 views
Does there exist a 3-connected, chordal graph which is not globally rigid?
The question is in the title! I know that a globally rigid graph is 3-connected and redundantly rigid, so my question could be rephrased as: "does there exist a graph which is 3-connected and chordal ...
19
votes
1answer
941 views
Why is there a unique hyperbolic simplex of largest area?
Why is there a unqiue ideal $n$-simplex in $\mathbb H^n$ with largest volume for $n\geq 3$?
For $n=3$, this is a standard calculation, and for larger dimensions is much harder (see Haagerup and ...
4
votes
2answers
341 views
Isostatic graphs and the Henneberg conjecture
I have been reading "Combinatorial Rigidity" by Graver, Servatius and Servatius and I am interested in their chapter on rigidity in dimension $\geq$ 3. I have two questions.
What is the current ...
3
votes
0answers
131 views
Show that duality functor is anti-monoidal
Let $\mathcal{C}$ be a right rigid (not strict) monoidal category with associativity constraint $\Phi$. Let $J_{UV}: U^*\otimes V^*\to (V\otimes U)^*$ the canonical isomorphism for every objects $U,V\...
3
votes
5answers
695 views
Is the following two-dimensional graph likely to be globally rigid?
Consider the two-dimensional non-planar graph $G$, with known topology and edge lengths $(r_1, r_2, ... r_N) \in R$, but unknown vertex coordinates. We further specify that:
All vertices within a ...
2
votes
0answers
213 views
Rigidity of Diophantine torus translations
Let $T_a:x\mapsto x+a$ be a Diophantine translation on the torus $\mathbb T^d$, $d>1$.
Let $h$ be some $C^1$ diffeomorphism of $\mathbb T^d$ such that
$$
g=h\circ T_a\circ h^{-1}
$$
is $C^\infty$. ...
