6 problems

Intrinsic characterization conjecture. $(\mathbb S^n,g)$ admits a smooth global isometric embedding

Let $X$ be a normed space with unit sphere $S_X$, and let $Y$ range over normed spaces with unit spheres $S_Y$. An isometric embedding $S_Y\to S_X$ is a distance-preserving map bet…

Let $\Sigma$ be a $2$-dimensional compact Riemannian manifold, possibly with boundary, with positive Gauss curvature. A $C^{1,\alpha}$ isometric embedding is an isometric embedding…

Let $\Sigma_a$ be the polar cap and let $\mathscr{I}^\alpha_1(\Sigma_a)$ denote the class of $C^{1,\alpha}$ isometric embeddings of $\Sigma_a$ into the relevant codimension-one Euc…

Bryant–Griffiths–Yang conjecture. The characteristic variety $\Sigma({\bf c})$ is not smooth in $\mathbb R^n\setminus\\{0\\}$ for every $n\ge 5$.

Let $n\geq 3$ and let $M^n$ be an $n$-dimensional $PL$-manifold. A PL isometric embedding is an embedding of $M^n$ into Euclidean space that preserves its piecewise-linear metric.…