85 problems

Let ${\mathbb H}^1$ be the three-dimensional sub-Riemannian Heisenberg group. A Pansu sphere is the surface obtained as the union of all sub-Riemannian geodesics of a fixed curvatu…

Sard conjecture. For every $x\in M$ and every integer $r\in[m,n-1]$, the set $\operatorname{Abn}^{r}_{\Delta}(x)$ has zero Lebesgue measure in $M$.

Let $\mathbb{G}$ be an $m$-dimensional Carnot group with horizontal distribution $\Delta=\operatorname{span}\{X_1,\ldots,X_k\}$, let $I=[0,1]$, and set $H=L^2(I,\mathbb{R}^k)$. For…

Cut-time and cut-locus conjecture. The cut time should be

Let $M$ be a smooth manifold of dimension $3$ equipped with a bracket-generating distribution $\Delta$, and for $x\in M$ let $\mathcal{X}^x_{\Delta}$ be the set of endpoints of sin…

Measure contraction conjecture. Carnot–Carathéodory spaces do not possess any type of measure contraction property.

The Heisenberg Bernstein conjecture. Then $\mathcal S$ should be a vertical plane $\widetilde P_0$.

Let $\mathbb H^1$ denote the first Heisenberg group. An intrinsic graph is a graph expressed using the intrinsic horizontal structure of $\mathbb H^1$, and stability refers to nonn…

Let a sub-Riemannian sphere be the metric sphere associated with a sub-Riemannian structure, and consider the general case with abnormal minimizers. The log-exp category is the fun…

Let an SR sphere mean a sphere for a sub-Riemannian structure, and call the case non-integrable when the associated geodesic dynamics is not integrable. Il'Yashenko's category is t…

Let $\beta$ be the parameter appearing in the non-integrable SR Martinet problem, and let an SR Martinet sphere denote the corresponding sub-Riemannian sphere. A function is pfaffi…

Let $\mathbb H^1$ be the first Heisenberg group. An intrinsic graph is a graph with respect to a horizontal direction, and stability means nonnegative second variation of the $H$-p…

Distance-realization property. Then the distance $\operatorname{dist}(P,S_R)$ is realized by the point

Let $(M,\alpha,g)$ be a compact contact Riemannian manifold, with Reeb vector field associated to the contact form $\alpha$, and let the sub-Laplacian be the sub-Riemannian Laplaci…

Bogoyavlensky's conjecture. If the Euler equation $\dot x_0=[x_0,A_0(x_0)]$ is completely integrable, then the extended system consisting of this equation together with the equatio…

Direct-type criterion. The geodesic $\gamma(t)$ is of direct-type if and only if

Asymptotic-velocity conjecture. The metric lines in $\mathbb{G}$ are precisely the sub-Riemannian geodesics $\gamma(t)$ parameterized by arc length for which there exists a unit…

Metric-line classification conjecture. The metric lines in $\mathcal{J}^k(\mathbb{R},\mathbb{R}^n)$ are precisely the line, homoclinic, and direct-type geodesics.

Cut-time conjecture. The cut time $t_{\operatorname{cut}}$ is identically

Let $\mathbb{H}^1$ denote the first Heisenberg group. A $C^1_{\mathbb{H}}$-regular 2-dimensional submanifold with boundary is a 2-dimensional submanifold with boundary whose regula…

Let $M$ be a manifold with a distribution $\xi$, and let $\Xi$ be the distribution on the group of contactomorphisms whose fiber at the identity is … A distribution is bracket gene…

Failure of the measure contraction property with strong Goh–Legendre geodesics. If there is a strong-Goh–Legendre geodesic $\gamma$, then $\mathsf{d}^2:M\times M\to\mathbb{R}_+$ fa…

A sub-Finsler Carnot group is a stratified, nilpotent Lie group equipped with a left-invariant distance. Its homogeneous dimension is denoted by $\mathcal{Q}$, and…

Let $N_{\mathrm{CE}}$ denote the curvature exponent of a step-two Carnot group, namely the least $N$ for which the measure contraction property $\mathrm{MCP}(0,N)$ holds. Nonintegr…

Let $N_{\mathrm{CE}}$ be the curvature exponent of a Carnot group, namely the least $N$ for which the measure contraction property $\mathrm{MCP}(0,N)$ holds. Let $N_0$ be the lower…