arXiv research

Study shows not all smooth paths are optimal in certain geometric structures.

New examples of sub-Riemannian structures satisfying Minimizing Sard conjecture found.

This paper is concerned with the study of the Monge optimal transport problem in sub-Riemannian manifolds where the cost is given by the square of the sub-Riemannian distance. Our aim is to extend previous results on existence and uniqueness of optimal transport maps to cases of sub-Riemannian structures which admit ma…

Based on the notion of dilatation structure arXiv:math/0608536, we give an intrinsic treatment to sub-riemannian geometry, started in the paper arXiv:0706.3644 . Here we prove that regular sub-riemannian manifolds admit dilatation structures. From the existence of normal frames proved by Bellaiche we deduce the rest of…

We provide the first known family of examples of integrable homogeneous sub-Riemannian structures admitting strictly abnormal geodesics. These examples were obtained through the analysis of the equivalence problem for sub-Riemannian Engel structures. We formulate a criterion of strict abnormality in terms of structure …

Study approximates sub-Riemannian structures with Riemannian metrics and analyzes spectral convergence.

Derives sub-Riemannian Ricci curvature for various manifolds.

We solve the local equivalence problem for sub-Riemannian structures on (2n + 1)-dimensional manifolds. We show that two sub-Riemannian structures are locally equivalent if and only if? their corresponding canonical linear connections are equivalent. When n = 1, these connections coincide with the generalized Tanaka-We…

Study on geodesics in a specific sub-Riemannian structure with two types of behavior.

New sub-Riemannian structures fail synthetic curvature bounds.

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…

The notion of a parallelizable distribution has been introduced and investigated. A non-integrable parallelizable distribution carries a natural sub-Riemannian structure. The geometry of this structure has been studied from the bi-viewpoint of absolute parallelism geometry and sub-Riemannian geometry. Two remarkable li…

Each sub-Riemannian geometry with bracket generating distribution enjoys a background structure determined by the distribution itself. At the same time, those geometries with constant sub-Riemannian symbols determine a unique Cartan connection leading to their principal invariants. We provide cohomological description …

We prove the $C^{1}$ regularity for a class of abnormal length-minimizers in rank $2$ sub-Riemannian structures. As a consequence of our result, all length-minimizers for rank $2$ sub-Riemannian structures of step up to $4$ are of class $C^{1}$.

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…

The paper defines a new connection on sub-Riemannian manifolds and explores conditions for almost quasi-Sasakian manifolds to be Einstein.

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

For a fat sub-Riemannian structure, we introduce three canonical Ricci curvatures in the sense of Agrachev-Zelenko-Li. Under appropriate bounds we prove comparison theorems for conjugate lengths, Bonnet-Myers type results and Laplacian comparison theorems for the intrinsic sub-Laplacian. As an application, we consider …

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,…

Reduced sub-Riemannian time on a specific group structure.

Sharp proof of sub-Riemannian length-minimizing curves being at least $C^2$

Characterizes metabelian distributions and geodesics in sub-Riemannian manifolds.

Harmonic maps studied in sub-Riemannian geometry for Lie groups.

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

Study holonomy in pseudo-Hermitian geometry structures.

The paper uses singularity theory to find normal forms for sub-Riemannian exponential maps.

Study magnetic flows on 3D contact sub-Riemannian manifolds using Rumin complex.

Maps on Sasakian manifolds limit to sub-Riemannian distance bounds.

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

Researchers find shortest paths on a special group structure.

New framework segments 3D scenes using neural algorithms and sub-Riemannian geometry.

In the present paper we consider generic Sub-Riemannian structures on the co-rank 1 non-holonomic vector distributions and introduce the associated canonical volume and ''horizontal'' area forms. As in the classical case, the Sub-Riemannian minimal surfaces can be defined as the critical points of the '`horizontal'' ar…

We compute the asymptotic expansion of the volume of small sub-Riemannian balls in a contact 3-dimensional manifold, and we express the first meaningful geometric coefficients in terms of geometric invariants of the sub-Riemannian structure

Introduces canonical connections for sub-Riemannian manifolds with constant symbol.

We prove that two-step analytic sub-Riemannian structures on a compact analytic manifold equipped with a smooth measure and Lipschitz Carnot groups satisfy measure contraction properties.

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

Maps commuting with sub-Laplacians on Carnot groups are conformal.

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…

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

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

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 …

Researchers find metric lines in SE(2) using Hamilton-Jacobi theory.

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

Small sub-Riemannian balls have diameter close to twice their radius.

The paper develops a non-transitive Cartan connection for sub-Riemannian manifolds.

We prove that in a class of non-equiregular sub-Riemannian manifolds corners are not length minimizing. This extends the results [4]. As an application of our main result we complete and simplify the analysis in [6], showing that in a 4-dimensional sub-Riemannian structure suggested by Agrachev and Gauthier all length-…

The paper constructs optimal sub-Riemannian geodesics in specific Carnot groups.

In this article we study the sub-Riemannian geometry of the spheres $S^{2n+1}$ and $S^{4n+3}$, arising from the principal $S^1-$bundle structure defined by the Hopf map and the principal $S^3-$bundle structure given by the quaternionic Hopf map respectively. The $S^1$ action leads to the classical contact geometry of $…