arXiv research

In the present paper we show that the geodesic flows of a sub-Riemannian metric and that of a Riemannian extension commute if and only if the extended metric is parallel with respect to a certain connection. This helps us to describe the geodesic flow of sub-Riemannian metrics on totally geodesic Riemannian submersions…

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

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

Study on integrability of geodesic flows on Heisenberg group.

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

Geodesics spiral around Reeb orbits in 3D contact manifolds.

Integrates magnetic geodesic and sub-Riemannian flows on Stefel variety, proving integrability and Lax presentations.

Study geodesic curves on Heisenberg group, classify them, and compute first step of quadrature.

The chaotic geodesic flow on a jet space is non-integrable.

In this paper, we define and study sub-Riemannian structures on Banach manifolds. We obtain extensions of the Chow-Rashevski theorem for exact controllability, and give conditions for the existence of a Hamiltonian geodesic flow despite the lack of a Pontryagin Maximum Principle in the infinite dimensional setting.

Study integrability of geodesic flow on specific Lie groups.

The paper introduces a new flow for Legendrian curves in Sasakian sub-Riemannian 3-manifolds.

Integrable geodesics found on special orthogonal group.

Study on homogeneous geodesics in sub-Riemannian geometry.

Study magnetic geodesics on odd spheres, computing critical energy values.

Study geodesics in sub-Riemannian manifolds, resolving open questions.

There are many equivalent definitions of Riemannian geodesics. They are naturally generalised to sub-Riemannian manifold, but become non-equivalent. We give a review of different definitions of geodesics of a sub-Riemannian manifold and interrelation between them. We recall three variational definitions of geodesics as…

We discuss the integrability of rank 2 sub-Riemannian structures on low-dimensional manifolds, and then prove that some structures of that type in dimension 6, 7 and 8 have a lot of symmetry but no integrals polynomial in momenta of low degrees, except for those coming from the Killing fields and the Hamiltonian, thus …

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

In this paper we prove a sub-Riemannian version of the classical Santaló formula: a result in integral geometry that describes the intrinsic Liouville measure on the unit cotangent bundle in terms of the geodesic flow. Our construction works under quite general assumptions, satisfied by any sub-Riemannian structure ass…

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

Consider a smooth manifold $M$ equipped with a bracket generating distribution $D$. Two sub-Riemannian metrics on $(M,D)$ are said to be projectively (resp. affinely) equivalent if they have the same geodesics up to reparameterization (resp. up to affine reparameterization). A sub-Riemannian metric $g$ is called rigid …

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

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

The abstract discusses how sub-Riemannian manifolds can have branching geodesics.

Characterizes metabelian distributions and geodesics in sub-Riemannian manifolds.

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…

The present paper is devoted to the problem of (local) geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on generic corank 1 distributions. Using Pontryagin Maximum Principle, we treat Riemannian and sub-Riemannian cases in an unified way and obtain some algebraic necessary conditions for the geodes…

We determine the lengths of all closed sub-Riemannian geodesics on the three-sphere. Our methods are elementary and allow us to avoid using explicit formulas for the sub-Riemannian geodesics.

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

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

This paper provides some partial regularity results for geodesics (i.e., isometric images of intervals) in arbitrary sub-Riemannian and sub-Finsler manifolds. Our strategy is to study infinitesimal and asymptotic properties of geodesics in Carnot groups equipped with arbitrary sub-Finsler metrics. We show that tangents…

Study random walks on sub-Riemannian manifolds using retractions.

Study on geodesics in Cartan group sub-Riemannian problem, proving conjugate time relation to Maxwell time.

New optimality conditions for sub-Riemannian geodesics derived.

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

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 …

If a piece of the contour of a picture is missing to the eye vision, then the brain tends to complete it using some kind of sub-Riemannian geodesics of the unit tangent bundle of the plane, R2xS1. These geodesics can be obtained by lifting extremal curves of a total curvature type energy in the plane. We completely sol…

We introduce a notion of geodesic curvature $k_ζ$ for a smooth horizontal curve $ζ$ in a three-dimensional contact sub-Riemannian manifold, measuring how much a horizontal curve is far from being a geodesic. We show that the geodesic curvature appears as the first corrective term in the Taylor expansion of the sub-Riem…

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

In this paper we describe the geodesics of a left-invariant sub-Riemannian metric on the three-dimensional solvable Lie group $SOLV^-$.

The unit sphere $\mathbb S^3$ can be identified with the unitary group SU(2). Under this identification the unit sphere can be considered as a non-commutative Lie group. The commutation relations for the vector fields of the corresponding Lie algebra define a 2-step sub-Riemannian manifold. We study sub-Riemannian geod…

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

Study shows only hyperplanes in Heisenberg groups have zero curvature.

Researchers find shortest paths on a special group structure.

Considering Riemannian submersions, we find necessary and sufficient conditions for when sub-Riemannian normal geodesics project to curves of constant first geodesic curvature or constant first and vanishing second geodesic curvatures. We describe a canonical extension of the sub-Riemannian metric and study geometric p…

In this paper we study geodesics of left-invariant sub-Riemannian metrics on SO(3) and almost-Riemannian metrics on $S^2$. These structures are connected with each other, and it is possible to use information about one of them to obtain results about another one. We give an explicit parameterization of sub-Riemannian g…

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