7 problems

Let $\mathrm{St}(n,p)$ be the Stiefel manifold, let $U,\widetilde{U}\in\mathrm{St}(n,p)$, and let $U_N,\widetilde{U}_N\in\mathrm{St}(n+N,p)$ be obtained by appending $N$ zero rows…

Let $n$ and $p$ be integers with $p\leq n$, let $\beta>0$ parametrize the Riemannian metric on the Stiefel manifold $\mathrm{St}(n,p)$, and consider the lower bound for the corresp…

Let $G$ be a nonempty connected graph. Write $V(G)$ for its vertex set and $d_G$ for its geodesic distance. Let $\mathfrak{MB}$ denote the class of metric spaces $(X,d)$ such that…

Let $\mathcal{M}$ be a Riemannian manifold, and let $\operatorname{Diff}_\text{c}(\mathcal{M})$ be its group of compactly supported diffeomorphisms equipped with a right-invariant…

Let $\mathcal{D}_\mu(\mathbb{T}^2)$ be the volumorphism group of the flat torus equipped with the $\dot{H}^{-1/2}$ metric. Its sectional curvature is unbounded of both signs. Micho…

Let $M$ be the underlying manifold and let $s>\frac12$. Consider the corresponding right-invariant Sobolev metric on the diffeomorphism group. Positivity conjecture. The geodesic d…

Let $M=\mathbb{R}$ and let $s=\frac12$. Consider the right-invariant Sobolev metric of order $s$ on $\operatorname{Diff}_c(M)$. Vanishing-distance conjecture. The geodesic distance…