All Questions
Tagged with rigidity riemannian-geometry
7 questions
3
votes
0
answers
57
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Examples of rigid open surfaces
In the celebrated book of Hilbert and Cohn-Vossen, the following sentence appears (p. 230):
Bending is impossible in the case of all closed convex surfaces, such as, for example, the ellipsoids. It is ...
- 797
0
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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 ...
- 63.9k
3
votes
1
answer
340
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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{...
- 108k
7
votes
2
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371
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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 ...
- 12.3k
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 $...
- 6,741
14
votes
2
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872
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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 $\...
- 108k
4
votes
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Conformal $L^p$ rigidity of Riemannian manifolds
$\newcommand{\M}{\mathcal{M}}$
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$\newcommand{\R}{\mathbb{R}}$
$\newcommand{\CO}[1]{\text{CO}(#1)}$
$\newcommand{\dist}{\operatorname{dist}}$
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