arXiv research

Complete classification of sub-Riemannian spaces with step and rank three.

Study compares sub-Riemannian curvature to optimal control variational problems.

We introduce length dilatation structures on metric spaces, tempered dilatation structures and coherent projections and explore the relations between these objects and the Radon-Nikodym property and Gamma-convergence of length functionals. Then we show that the main properties of sub-riemannian spaces can be obtained f…

Constructs stochastic processes on sub-Riemannian manifolds using Cartan connections.

Paper derives sub-Riemannian versions of Kastler-Kalau-Walze and Dabrowski-Sitarz-Zalecki theorems for twisted BCV spaces.

We prove sectional and Ricci-type comparison theorems for the existence of conjugate points along sub-Riemannian geodesics. In order to do that, we regard sub-Riemannian structures as a special kind of variational problems. In this setting, we identify a class of models, namely linear quadratic optimal control systems,…

Unified study of Riemannian and sub-Riemannian geometries with synthetic Ricci curvature bounds.

Sub-Riemannian cubics are a generalisation of Riemannian cubics to a sub-Riemannian manifold. Cubics are curves which minimise the integral of the norm squared of the covariant acceleration. Sub-Riemannian cubics are cubics which are restricted to move in a horizontal subspace of the tangent space. When the sub-Riemann…

Study of intrinsic sub-Laplacian for hypersurfaces in contact sub-Riemannian manifolds.

Developed a sub-Riemannian version of Minkowski problem in Heisenberg groups.

New sub-Riemannian spaces with boundary meet curvature-dimension condition.

We prove a version of Gauss-Bonnet theorem in sub-Riemannian Heisenberg space $H^1$. The sub-Riemannian distance makes $H^1$ a metric space and consenquently with a spherical Hausdorff measure. Using this measure, we define a Gaussian curvature at points of a surface S where the sub-Riemannian distribution is transvers…

Generalizes Connes's tangent groupoid for sub-Riemannian geometry.

Proves Steiner and tube formulae for 3D contact sub-Riemannian surfaces.

The study establishes inequalities on path space for sub-Riemannian manifolds.

The study proves sub-Riemannian manifolds cannot satisfy $\mathrm{CD}$ conditions unless they are Riemannian.

Extended orbit model theory for shape analysis using graded group action framework.

We extend Agrachev-Brockett-Jurjdevic's solution to normal sub-Riemannian geodesics.

Sub-Riemannian Geometry is proved to play an important role in many applications, e.g., Mathematical Physics and Control Theory. The simplest example of sub-Riemannian structure is provided by the 3-D Heisenberg group. Sub-Riemannian Geometry enjoys major differences from the Riemannian being a generalisation of the la…

Sub-Riemannian spectral distance defined using eigenfunctions of sub-Laplacian

This study classifies metric lines in jet space.

Geodesics spiral around Reeb orbits in 3D contact manifolds.

This paper classifies Legendre singularities of sub-Riemannian geodesics on surfaces.

Study sub-Riemannian surfaces in contact manifolds, proving a Gauss-Bonnet theorem.

Proves sub-Riemannian manifolds are infinitesimally Hilbertian with arbitrary measures.

Study sub-Riemannian geodesics on a Heisenberg 3D nil-manifold.

Study on homogeneous geodesics in sub-Riemannian geometry.

Length spectra for Riemannian metrics are well studied, while sub-Riemannian length spectra have been largely unexplored. Here we give the length spectrum for a canonical sub-Riemannian structure attached to any compact Lie group by restricting its Killing form to the sum of the root spaces. Surprisingly, the shortest …

The paper finds two types of metric lines in curve spaces.

The study finds instability conditions for specific surfaces in sub-Riemannian 3-space forms.

Paper proves inequalities for forms on sub-Riemannian manifolds.

We consider sub-Riemannian spaces admitting an isometry group that is maximal in the sense that any linear isometry between the horizontal tangent spaces is realized by a global isometry. We will show that these spaces have a canonical choice of partial connection on their horizontal bundle, which is determined by isom…

The exponential map fails to be injective near critical points in sub-Riemannian geometry.

Derivatives of sub-Riemannian geodesics are always $L_p$-Hölder continuous.

Gromov proposed to extract the (differential) geometric content of a sub-riemannian space exclusively from its Carnot-Carathéodory distance. One of the most striking features of a regular sub-riemannian space is that it has at any point a metric tangent space with the algebraic structure of a Carnot group, hence a homo…

Sub-Riemannian structures on odd-dimensional spheres respecting the Hopf fibration naturally appear in quantum mechanics. We study the curvature maps for such a sub-Riemannian structure and express them using the Riemannian curvature tensor of the Fubini-Study metric of the complex projective space and the curvature fo…

New insights into integrability and rectifiability in sub-Riemannian geometry.

Paper studies third order open mapping in sub-Riemannian geometry.

We study the classification of area-stationary and stable $C^2$ regular surfaces in the space of the rigid motions of the Minkowski plane E(1,1), equipped with its sub-Riemannian structure. We construct examples of area-stationary surfaces that are not foliated by sub-Riemannian geodesics. We also prove that there exis…

For constant mean curvature surfaces of class $C^2$ immersed inside Sasakian sub-Riemannian 3-manifolds we obtain a formula for the second derivative of the area which involves horizontal analytical terms, the Webster scalar curvature of the ambient manifold, and the extrinsic shape of the surface. Then we prove classi…

The authors found geodesics, shortest arcs, cut loci, and conjugate sets for left-invariant sub-Riemannian matric on the Lie group $SL(2)$, which is right-invariant relative to the Lie subgroup $SO(2)\subset SL(2)$ (in other words, for invariant sub-Riemannian metric on weakly symmetric space $(SL(2)\times SO(2))/SO(2)…

Holonomy groups of K-contact sub-Riemannian manifolds are isomorphic.

Researchers find shortest paths on a special group structure.

New sub-Riemannian structures fail synthetic curvature bounds.

The curvature discussed in this paper is a rather far going generalization of the Riemannian sectional curvature. We define it for a wide class of optimal control problems: a unified framework including geometric structures such as Riemannian, sub-Riemannian, Finsler and sub-Finsler structures; a special attention is p…

New method for sampling diffusion bridges on sub-Riemannian manifolds.

A surface of constant mean curvature (CMC) equal to $H$ in a sub-Riemannian $3$-manifold is strongly stable if it minimizes the functional $\text{area}+2H\,\text{volume}$ up to second order. In this paper we obtain some criteria ensuring strong stability of surfaces in Sasakian $3$-manifolds. We also produce new exampl…

Let $M$ be a complete Sasakian sub-Riemannian $3$-manifold of constant Webster scalar curvature $κ$. For any point $p\in M$ and any number $λ\in\mathbb{R}$ with $λ^2+κ>0$, we show existence of a $C^2$ spherical surface $\mathcal{S}_λ(p)$ immersed in $M$ with constant mean curvature $λ$. Our construction recovers in par…