Questions tagged [rigidity]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
2 votes
0 answers
134 views

Counterexample to mostow rigidity theorem

I am looking for an example of $M$ and $N$ two orientable hyperbolic complete without boundary 3-manifolds ( with infinite hyperbolic volume) such that $\pi_{1}M\cong \pi_{1}N$ but $M$ is not ...
user avatar
  • 321
2 votes
0 answers
98 views

$l$-adic rigidity for Milnor $K$-theory

Given a local henselian ring $A$ with the maximal ideal $m$, does the quotient map $A\mapsto A/m$ induce isomorphisms on $l$-adic Milnor $K$-theories? ($K_n^M(R)\otimes \mathbb{Z}_l$, where $l$ is an ...
user avatar
  • 4,926
7 votes
2 answers
289 views

Constant Gaussian curvature disks

This question has also been posted on MSE, but maybe here is the right place to post it. Is it true that if $D$ is a Riemannian $2$-disk having constant Gaussian curvature equal to $1$ and whose ...
user avatar
2 votes
0 answers
65 views

Infinitesimal rigidity vs local rigidity of isometrically immersed riemannian manifolds

I was reading the nice survey on rigidity, focusing on tensegrities by Connelly, and I'd like to know the status and feedback about a question he asks: A theorem by Gluck and this work of Connelly, ...
user avatar
  • 2,001
10 votes
0 answers
271 views

How to correctly state Cauchy's rigidity theorem?

Cauchy's rigidity theorem is usually cites briefly as Any two (convex, 3-dimensional) polyhedra with pairwise congruent faces are themselves congruent. As a more formal generalization to general ...
user avatar
  • 9,481
4 votes
2 answers
238 views

Forbidden minors of a graph with treewidth at most 4

I am interested in the graphs with treewidth 5 because of its relationship with realization dimension of a graph (See here). In this PhD thesis, 75 lists of minimal forbidden minors of a graph with ...
user avatar
2 votes
0 answers
77 views

Are a map with constant singular values and its inverse always conjugate through isometries?

Let $U \subseteq \mathbb R^2$ be open, connected and bounded, and let $0<\sigma_1<\sigma_2$ satisfy $\sigma_1 \sigma_2=1$. Suppose that $f:U \to U$ is a diffeomorphism whose singular values (of $...
user avatar
  • 6,286
3 votes
1 answer
154 views

Are "strongly finite dimensional" homotopy invariant sheaves with transfers (locally) constant?

Let $k$ be an algebraically closed field. Let $S$ be a homotopy invariant $\mathbb{Q}$-linear sheaf with transfers in the sense of Voevodsky–Suslin, and assume that the dimension of $S(U)$ (over $\...
user avatar
12 votes
2 answers
787 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 $\...
user avatar
  • 6,286
5 votes
3 answers
561 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 ...
user avatar
  • 9,481
4 votes
1 answer
241 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 ...
user avatar
6 votes
1 answer
317 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 $c(gg',x)=c(g,g'x)c(g',x)$. ...
user avatar
  • 123
4 votes
0 answers
78 views

Conformal $L^p$ rigidity of Riemannian manifolds

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\CO}[1]{\text{CO}(#1)}$ $\newcommand{\dist}{\operatorname{dist}}$ $\newcommand{\g}{\mathfrak{g}}...
user avatar
  • 6,286
9 votes
1 answer
801 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 ...
user avatar
26 votes
3 answers
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 ...
user avatar
  • 37.1k
2 votes
1 answer
151 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, ...
user avatar
  • 21
5 votes
2 answers
157 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 ...
user avatar
  • 12.4k
11 votes
1 answer
513 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 ...
user avatar
  • 1,002
4 votes
1 answer
172 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 ...
user avatar
  • 2,131
10 votes
1 answer
484 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)...
user avatar
  • 2,462
3 votes
0 answers
148 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 ...
user avatar
11 votes
0 answers
650 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 ...
user avatar
1 vote
1 answer
190 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 ...
user avatar
  • 399
20 votes
1 answer
1k 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 ...
user avatar
  • 17.5k
4 votes
2 answers
364 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 ...
user avatar
4 votes
1 answer
179 views

Show that duality functor is anti-monoidal

Let $\mathcal{C}$ be a right rigid (not strict) monoidal category with associativity constraint $\Phi$. Let $J_{U,V}: U^*\otimes V^*\to (V\otimes U)^*$ be the canonical isomorphism for every objects $...
user avatar
  • 41
3 votes
5 answers
769 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 ...
user avatar
  • 309
2 votes
0 answers
223 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 $C^\infty$. ...
user avatar