arXiv research

Study periodic geodesics on contact 3D manifolds, proving existence and precise properties.

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

Geodesics spiral around Reeb orbits in 3D contact manifolds.

Study Schrödinger evolution on surfaces in 3D contact sub-Riemannian manifolds.

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

Quantifies tightness in 3D contact manifolds using sub-Riemannian metrics.

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

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

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

In this paper we study the small time asymptotics for the heat kernel on a sub-Riemannian manifold, using a perturbative approach. We then explicitly compute, in the case of a 3D contact structure, the first two coefficients of the small time asymptotics expansion of the heat kernel on the diagonal, expressing them in …

Unified interpretation of sub-Riemannian Gauss-Bonnet theorem for surfaces in 3D contact manifolds.

In this paper, we extend and complete the classification of the generic singularities of the 3D-contact sub-Riemmanian conjugate locus in a neighbourhood of the origin.

In this paper we provide the small-time heat kernel asymptotics at the cut locus in three relevant cases: generic low-dimensional Riemannian manifolds, generic 3D contact sub-Riemannian manifolds (close to the starting point) and generic 4D quasi-contact sub-Riemannian manifolds (close to a generic starting point). As …

Study on curvatures of surfaces in specific Lie groups.

In this paper a conformal classification of three dimensional left-invariant sub-Riemannian contact structures is carried out; in particular we will prove the following dichotomy: either a structure is locally conformal to the Heisenberg group $\mathbb H_3$, or its conformal classification coincides with the metric one…

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

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

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

We prove existence of regions minimizing perimeter under a volume constraint in contact sub-Riemannian manifolds such that their quotient by the group of contact transformations preserving the sub-Riemannian metric is compact.

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

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…

Study stochastic processes on surfaces in contact sub-Riemannian manifolds using Riemannian approximations.

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

Paper proves inequalities for forms on sub-Riemannian manifolds.

Study contact 3-manifolds using sub-Riemannian geometry, proving new properties of their Lipschitz homotopy groups.

On H-type sub-Riemannian manifolds we establish sub-Hessian and sub-Laplacian comparison theorems which are uniform for a family of approximating Riemannian metrics converging to the sub-Riemannian one. We also prove a sharp sub-Riemannian Bonnet-Myers theorem that extends to this general setting results previously pro…

The paper proves a decomposition theorem for forms on sub-Riemannian contact manifolds.

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

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

This is the first paper of a series in which we plan to study spectral asymptotics for sub-Riemannian Laplacians and to extend results that are classical in the Riemannian case concerning Weyl measures, quantum limits, quantum ergodicity, quasi-modes, trace formulae.Even if hypoelliptic operators have been well studied…

We study the sub-Riemannian exponential for contact distributions on manifolds of dimension greater or equal to 5. We compute an approximation of the sub-Riemannian Hamiltonian flow and show that the conjugate time can have multiplicity 2 in this case. We obtain an approximation of the first conjugate locus for small r…

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…

In this paper we consider surfaces of class $C^1$ with continuous prescribed mean curvature in a three-dimensional contact sub-Riemannian manifold and prove that their characteristic curves are of class $C^2$. This regularity result also holds for critical points of the sub-Riemannian perimeter under a volume constrain…

We construct a canonically defined affine connection in sub-Riemannian contact geometry. Our method mimics that of the Levi-Civita connection in Riemannian geometry. We compare it with the Tanaka-Webster connection in the three-dimensional case.

The paper explores conditions for constructing and extending infinitesimal isometries on special sub-Riemannian manifolds.

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

We prove a Bonnet-Myers type theorem for quaternionic contact manifolds of dimension bigger than 7. If the manifold is complete with respect to the natural sub-Riemannian distance and satisfies a natural Ricci-type bound expressed in terms of derivatives up to the third order of the fundamental tensors, then the manifo…

The geodesics for a sub-Riemannian metric on a three-dimensional contact manifold $M$ form a 1-parameter family of curves along each contact direction. However, a collection of such contact curves on $M$, locally equivalent to the solutions of a fourth-order ODE, are the geodesics of a sub-Riemannian metric only if a s…

We obtain some results on symmetries of sub-Riemannian surfaces. In case of contact sub-Riemannian surface we base on invariants found by Hughen \cite{Hughen}. Using these invariants, we find conditions under which a sub-Riemannian surface does not admit symmetries. If a surface admits symmetries, we show how invariant…

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

We study some sub-Riemannian objects (such as horizontal connectivity, horizontal connection, horizontal tangent plane, horizontal mean curvature) in hypersurfaces of sub-Riemannian manifolds. We prove that if a connected hypersurface in a contact manifold of dimension more than three is noncharacteristic or with isola…

New proof of wave trace formula for 3D-contact manifolds.

Notes on sub-Riemannian geometry equivalence problem.

3D contact manifolds have optimal higher systolic ratios.

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

In this paper we consider a set $E\subsetΩ$ with prescribed mean curvature $f\in C(Ω)$ and Euclidean Lipschitz boundary $\partial E=Σ$ inside a three-dimensional contact sub-Riemannian manifold $M$. We prove that if $Σ$ is locally a regular intrinsic graph, the characteristic curves are of class $C^2$. The result is sh…

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 study eigenvalue problems for intrinsic sub-Laplacians on regular sub-Riemannian manifolds. We prove upper bounds for sub-Laplacian eigenvalues $λ_k$ of conformal sub-Riemannian metrics that are asymptotically sharp as $k\to +\infty$. For Sasakian manifolds with a lower Ricci curvature bound, and more generally, for…