arXiv research

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

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

Researchers define geodesic curvature for 3D sub-Riemannian curves, improving distance calculations.

Study on surfaces of genus g≥1 in 3D contact sub-Riemannian manifolds, proving finiteness or infiniteness of induced distance.

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

In this paper we study global distance estimates and uniform local volume estimates in a large class of sub-Riemannian manifolds. Our main device is the generalized curvature dimension inequality introduced by the first and the third author in \cite{BG1} and its use to obtain sharp inequalities for solutions of the sub…

We prove global estimates for the sub-Riemannian distance of CR Sasakian manifolds with non negative horizontal Webster-Tanaka Ricci curvature. In particular, in this setting, large sub-Riemannian balls are comparable to Riemannian balls.

Study radial processes in sub-Riemannian Brownian motions, proving stochastic completeness and eigenvalue estimates.

Study optimal transport in 4D sub-Riemannian spaces with many singular geodesics.

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

Study investigates induced geometry on surfaces in 3D contact manifolds.

Derives sub-Riemannian Ricci curvature for various manifolds.

We compare different notions of curvature on contact sub-Riemannian manifolds. In particular we introduce canonical curvatures as the coefficients of the sub-Riemannian Jacobi equation. The main result is that all these coefficients are encoded in the asymptotic expansion of the horizontal derivatives of the sub-Rieman…

We find a different approach to define convex functions in the sub-Riemannian setting. A function on a sub-Riemannian manifold is nonholonomically geodesic convex if its restriction to any nonholonomic (straightest) geodesic is convex. In the case of Carnot groups, this definition coincides with that by Danniell-Garofa…

The study proves a theorem for quaternionic contact manifolds, showing they are compact under certain conditions.

In this paper we study the main geometric properties of the Carnot-Carathéodory (abbreviated CC) distance $\dc$ in the setting of $k$-step sub-Riemannian Carnot groups from many different points of view. An extensive study of the so-called normal CC-geodesics is given. We state and prove some related variational formul…

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

Study on diffusion in non-complete sub-Riemannian manifolds with specific conditions.

Study geodesic curvature in Heisenberg group, interpreting it as distance correction.

Reduced sub-Riemannian time on a specific group structure.

Study develops geodesic theory for foliations, proving Laplacian comparison theorems.

In infinite dimensional Heisenberg group, degenerate distances linked to unbounded curvature.

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

Study on distances and neighborhoods in Heisenberg groups, proving regularity and curvature.

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…

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…

Study exact formula and geodesics for Carnot-Carathéodory distance on 2-step groups.

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

Study geodesics and shortest arcs on Lie groups with specific metrics.

A left-invariant sub-Riemannian metric $d$ on the shortened Lorentz group $SO_0(2,1)$ under the condition that $d$ is right-invariant relative to the orthogonal Lie subgroup $1\otimes SO(2)$ is studied. The distance between arbitrary two elements, the cut locus (as the union of the subgroup $1\otimes SO(2)$ with the an…

We use a Riemannnian approximation scheme to define a notion of $\textit{sub-Riemannian Gaussian curvature}$ for a Euclidean $C^{2}$-smooth surface in the Heisenberg group $\mathbb{H}$ away from characteristic points, and a notion of $\textit{sub-Riemannian signed geodesic curvature}$ for Euclidean $C^{2}$-smooth curve…

The authors compute distances between arbitrary elements of Lie groups SU(2) and SO(3) for special left-invariant sub-Riemannian metrics $ρ$ and $d$. To compute distances for the second metric, we essentially use the fact that canonical two-sheeted covering epimorphism $Ω$ of the Lie group SU(2) onto the Lie group SO(3…

Our main aim is to present a geometrically meaningful formula for the fundamental solutions to a second order sub-elliptic differential equation and to the heat equation associated with a sub-elliptic operator in the sub-Riemannian geometry on the unit sphere $\mathbb S^3$. Our method is based on the Hamiltonian approa…

New sub-Riemannian structures fail synthetic curvature bounds.

Study validates Michor-Mumford conjecture in infinite dimensional Hilbertian H-type groups.

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…

We study sub-Riemannian and sub-Lorentzian geometry on the Lie group $\SU(1,1)$ and on its universal cover $\CSU(1,1)$. In the sub-Riemannian case we find the distance function and completely describe sub-Riemannian geodesics on both $\SU(1,1)$ and $\CSU(1,1)$, connecting two fixed points. In particular, we prove that …

Ideal sub-Riemannian manifolds support interpolation inequalities for optimal transport.

The paper studies sub-Riemannian geometry and proves a Weyl's invariance result for Heisenberg groups.

A new snake model improves segmentation of SEM images.

Study heat content in sub-Riemannian structures, proving asymptotic series existence and coefficients.

Measure contraction properties are generalizations of the notion of Ricci curvature lower bounds in Riemannian geometry to more general metric measure spaces. In this paper, we give sufficient conditions for a Sasakian manifold equipped with a natural sub-Riemannian distance to satisfy these properties. Moreover, the s…

We study mappings on sub-Riemannian manifolds which are quasi-regular with respect to the Carnot-Caratheodory distances and discuss several related notions. On H-type Carnot groups, quasiregular mappings have been introduced earlier using an analytic definition, but so far, a good working definition in the same spirit …

The paper introduces a method for dimension reduction using sub-Riemannian geometry.

Sharp estimates on 2-step nilpotent Lie groups' metrics and cones.

This is a review of explicit computations of Connes distance in noncommutative geometry, covering finite dimensional spectral triples, almost-commutative geometries, and spectral triples on the algebra of compact operators. Several applications to physics are covered, like the metric interpretation of the Higgs field, …

We study a transformation of metric measure spaces introduced by Gigli and Mantegazza consisting in replacing the original distance with the length distance induced by the transport distance between heat kernel measures. We study the smoothing effect of this procedure in two important examples. Firstly, we show that in…

The Heston model is a popular stock price model with stochastic volatility that has found numerous applications in practice. In the present paper, we study the Riemannian distance function associated with the Heston model and obtain explicit formulas for this function using geometrical and analytical methods. Geometric…