All Questions

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 ...

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{...

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 $...

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 $\...