# Questions tagged [rigidity]

The rigidity tag has no usage guidance.

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### Is a convex polyhedron determined by its edge lengths and angular defects?

Let's consider 3-dimensional convex polyhedra $P\subset\Bbb R^3$.
The angular defect at a vertex $v$ is $2\pi$ minus the sum of the interior angles of the incident faces at $v$.
Question:
Is a ...

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### How many rigid 4-regular graphs are there?

I am interested in any formulas for the number of globally rigid 4-regular graphs, or Laman graphs of degree at most 4, on $n$ vertices. The bound can be for graphs with labeled or unlabeled vertices.

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### Tannaka duality for Hopf algebroids

Setting. Let $k$ be a field, $A$ a finite-dimensional $k$-algebra, and $H$ a Hopf algebroid over $A$ with invertible antipode. Denote by $\operatorname{mod}(H)$ the category of finite-dimensional ...

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### The rigidity of $2$-dim sphere with constant sectional curvature in $\mathbb{R}^n$ for $n> 3$

If there is a smooth isometric embedding $f: (S^2, g)\rightarrow \mathbb{R}^n$, where $(S^2, g)$ is a sphere with Riemannian metric such that the corresponding sectional curvature is equal to $1$, and ...

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### Ways of proving that a framework is locally rigid

Given a (bar-and-joint) framework/linkage, I would like to know what are possible ways of showing that the framework is locally rigid. Also, what is known about the computational complexity of ...

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### Frameworks in general position that are locally rigid but not infinitesimally rigid

The classical theorem of Asimow and Roth says that for a generic framework (i.e., coordinates of the nodes are algebraically independent), local rigidity and infinitesimal rigidity are equivalent. I ...

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### Which polytopes can be folded to an edge?

While playing with bar-and-joint linkages, I noticed that the skeleton of a regular 3-dimensional cube can be folded to a single edge (this can be achieved by first flexing the cube to bring it to a ...

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### Regarding rigid graphs in the plane

Quoting from the book (page 272) Graphs and Geometry by Lovasz, we have the following theorems regarding the characterization of rigid graphs in the pane.
Theorem 1: A graph $G$ is rigid in the plane ...

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### Is the dodecahedron flexible (as a polytope with fixed edge-lengths)?

Consider the (regular) dodecahedron $D\subset\Bbb R^3$. I want to continuously deform it so that throughout the deformation
it stays a convex polytope,
it stays a combinatorial dodecahedron (i.e. its ...

3
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1
answer

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### Shrinking a disk with fixed differential

Consider mappings $f$ from $\mathbb{R}^2$ to $\mathbb{R}^2$ with differential
\begin{align}
\mathsf{d} f= \begin{pmatrix}
\cos\psi(x) &\cos\phi(y) \\
\sin \psi(x)& \sin\phi(y)
\end{...

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### Right unitor in star-autonomous categories

1.Context
Let $(C, \otimes, I, a, l,r)$ be a monoidal category. Here $r$ denotes the right unitor.
Suppose $S: C^{op} \xrightarrow{\sim} C$ is an equivalence of categories with inverse $S’$. Assume ...

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### Cartesian monoidal star-autonomous categories

Disclaimer: This is a crosspost (see MathStackexchange). Apologies if cross-posting is frowned upon. However, it seems that on Stackexchange there are not many people familiar with star-autonomous ...

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262
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### 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 ...

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### $l$-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 ...

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### 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 ...

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### 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, ...

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523
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### 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 ...

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### Forbidden minors of a graph with treewidth at most 4

I am interested in the graphs with treewidth 5 because of their relationship with the realization dimension of a graph (see here).
In this PhD thesis, 75 minimal forbidden minors of graphs with ...

2
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### 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 $...

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1
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### 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 $\...

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2
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857
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### 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 $\...

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### 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 ...

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305
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### 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 ...

6
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418
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### 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 $c(gg',x)=c(g,g'x)c(g',x)$. ...

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### 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}}...

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851
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### 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 ...

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3
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### 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 ...

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### 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, ...

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### 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 ...

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567
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### 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

1
answer

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### 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 ...

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### 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)...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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1
answer

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### 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 $...

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### 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 ...

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### 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 $C^\infty$. ...