Conjugate points in nilpotent sub-Riemannian problem on the Engel group

The left-invariant sub-Riemannian problem on the Engel group is considered. This problem is very important as nilpotent approximation of nonholonomic systems in four-dimensional space with two-dimensional control, for instance of a system which describes motion of mobile robot with a trailer. We study local optimality of extremal trajectories and estimate conjugate time in this article.     

Cut Locus in the Sub-Riemannian Problem on Engel Group

Doklady Mathematics - This paper considers the left-invariant sub-Riemannian problem on the Engel group. The stratification of the cut locus and the structure of optimal synthesis are described.     

Hypoelliptic Heat Kernel Over 3-Step Nilpotent Lie Groups

In this paper, we provide explicitly the connection between the hypoelliptic heat kernel for some 3-step sub-Riemannian manifolds and the quartic oscillator. We study the left-invariant sub-Riemannian structure on two nilpotent Lie groups, namely, the (2,3,4) group (called the Engel group) and the (2,3,5) group (called the Cartan group or the generalized Dido problem). Our main technique is noncommutative Fourier analysis, which permits us to transform the hypoelliptic heat equation into a one-dimensional heat equation with a quartic potential.     

Extremal Controls in the Sub-Riemannian Problem on the Group of Motions of Euclidean Space

For the sub-Riemannian problem on the group of motions of Euclidean space we present explicit formulas for extremal controls in the special case where one of the initial momenta is fixed.     

Maxwell Strata and Cut Locus in the Sub-Riemannian Problem on the Engel Group

We consider the nilpotent left-invariant sub-Riemannian structure on the Engel group. This structure gives a fundamental local approximation of a generic rank 2 sub-Riemannian structure on a 4-manifold near a generic point (in particular, of the kinematic models of a car with a trailer). On the other hand, this is the simplest sub-Riemannian structure of step three. We describe the global structure of the cut locus (the set of points where geodesics lose their global optimality), the Maxwell set (the set of points that admit more than one minimizer), and the intersection of the cut locus with the caustic (the set of conjugate points along all geodesics). The group of symmetries of the cut locus is described: it is generated by a one-parameter group of dilations R₊ and a discrete group of reflections Z₂ × Z₂ × Z₂. The cut locus admits a stratification with 6 three-dimensional strata, 12 two-dimensional strata, and 2 one-dimensional strata. Three-dimensional strata of the cut locus are Maxwell strata of multiplicity 2 (for each point there are 2 minimizers). Two-dimensional strata of the cut locus consist of conjugate points. Finally, one-dimensional strata are Maxwell strata of infinite multiplicity, they consist of conjugate points as well. Projections of sub-Riemannian geodesics to the 2-dimensional plane of the distribution are Euler elasticae. For each point of the cut locus, we describe the Euler elasticae corresponding to minimizers coming to this point. Finally, we describe the structure of the optimal synthesis, i. e., the set of minimizers for each terminal point in the Engel group.     

Short-time asymptotics of heat kernels of hypoelliptic Laplacians on unimodular Lie groups

We consider the problem of computing heat kernel small-time asymptotics for hypoelliptic Laplacians associated to left-invariant sub-Riemannian structures on unimodular Lie groups of type I. We use the non-commutative Fourier transform of the Lie group together with perturbation theory for semigroups of operators in deriving these asymptotics. We illustrate our approach on the example of the Heisenberg group, and, as an application, we compute the short-time behaviour of the hypoelliptic heat kernel on the step 3 nilpotent Cartan and Engel groups, for which no closed-form expression for the hypoelliptic heat kernel is yet known.     

Sub-Riemannian Balls on the Heisenberg Groups: An Invariant Volume

The Popp measure of a sub-Riemannian ball is calculated for a left-invariant sub-Riemannian structure on the Heisenberg group.     

Liouville nonintegrability of sub-Riemannian problems on free Carnot groups of step 4

One of the main approaches to the study of the Carnot–Carathéodory metrics is the Mitchell–Gromov nilpotent approximation theorem, which reduces the consideration of a neighborhood of a regular point to the study of the left-invariant sub-Riemannian problem on the corresponding Carnot group. A detailed analysis of sub-Riemannian extremals is usually based on the explicit integration of the Hamiltonian system of Pontryagin’s maximum principle. In this paper, the Liouville nonintegrability of this system for left-invariant sub-Riemannian problems on free Carnot groups of step 4 and higher is proved.     

Measures of transverse paths in sub-Riemannian geometry

We define a class of lengths of paths in a sub-Riemannian manifold. It includes the length of horizontal paths but also measures the length of transverse paths. It is obtained by integrating an infinitesimal measure which generalizes the norm on the tangent space. This requires the definition and the study of the metric tangent space (in Gromov's sense). As an example, we compute those measures in the case of contact sub-Riemannian manifolds.     

Asymptotics of Extremal Curves in the Ball Rolling Problem on the Plane

In the present paper, we study an optimal sphere rolling problem on the plane (without slew and slip) with predefined boundary-value conditions. To solve it, we use methods from the optimal control theory. The controlled system for sphere orientation is represented via the rotation quaternion. Asymptotics of extremal paths on a sphere rolling along small-amplitude sine waves is found.     

Mass transportation on sub-Riemannian structures of rank two in dimension four

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 many singular minimizing geodesics. We treat here the case of sub-Riemannian structures of rank two in dimension four.     

Conformal mappings and CR mappings on the Engel group

We show that conformal mappings between the Engel groups are CR or anti-CR mappings. This reduces the determination of conformal mappings to a problem in the theory of several complex analysis. The result about the group of CR automorphisms is used to determine the identity component of the group of conformal mappings on the Engel group.     

The normal sub-Riemannian geodesic flow on E(2) generated by a left-invariant metric and a right-invariant distribution

We consider a sub-Riemannian problem on the three-dimensional solvable Lie group E(2) endowed with a left-invariant metric and a right-invariant distribution. The problem is based on construction of a Hamilton structure for the given metric by the Pontryagin maximum principle.     

On the Dido Problem and Plane Isoperimetric Problems

This paper is a continuation of a series of papers, dealing with contact sub-Riemannian metrics on R³. We study the special case of contact metrics that correspond to isoperimetric problems on the plane. The purpose is to understand the nature of the corresponding optimal synthesis, at least locally. It is equivalent to studying the associated sub-Riemannian spheres of small radius. It appears that the case of generic isoperimetric problems falls down in the category of generic sub-Riemannian metrics that we studied in our previous papers (although, there is a certain symmetry). Thanks to the classification of spheres, conjugate-loci and cut-loci, done in those papers, we conclude immediately. On the contrary, for the Dido problem on a 2-d Riemannian manifold (i.e. the problem of minimizing length, for a prescribed area), these results do not apply. Therefore, we study in details this special case, for which we solve the problem generically (again, for generic cases, we compute the conjugate loci, cut loci, and the shape of small sub-Riemannian spheres, with their singularities). In an addendum, we say a few words about: (1) the singularities that can appear in general for the Dido problem, and (2) the motion of particles in a nonvanishing constant magnetic field.     

The geodesic flow of a sub-Riemannian metric on a solvable Lie group

We consider the sub-Riemannian problem on the three-dimensional solvable Lie group SOLV⁺. The problem is based on constructing a Hamiltonian structure for a given metric by the Pontryagin Maximum Principle.     

Parallel Algorithm and Software for Image Inpainting via Sub-Riemannian Minimizers on the Group of Rototranslations

The paper is devoted to an approach for image inpainting developed on the basis of neurogeometry of vision and sub-Riemannian geometry. Inpainting is realized by completing damaged isophotes (level lines of brightness) by optimal curves for the left-invariant sub-Riemannian problem on the group of rototranslations (motions) of a plane SE(2). The approach is considered as anthropomorphic inpainting since these curves satisfy the variational principle discovered by neurogeometry of vision. A parallel algorithm and software to restore monochrome binary or halftone images represented as series of isophotes were developed. The approach and the algorithm for computation of completing arcs are presented in detail.     

A note on ill-posedness of the Cauchy problem for Heisenberg wave maps

We introduce a notion of wave maps with a target in the sub-Riemannian Heisenberg group and study their relation with Riemannian wave maps with range in Lagrangian submanifolds. As an application we establish existence and eventually ill-posedness of the corresponding Cauchy problem.

     

Sub-Riemannian Geodesics on the 3-D Sphere

The unit sphere 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 geodesics on this sub-Riemannian manifold making use of the Hamiltonian formalism and solving the corresponding Hamiltonian system. The first author is partially supported by a research grant from the United State Army Research Office and a Hong Kong RGC competitive earmarked research grant #600607. The second and the third authors have been supported by the grant of the Norwegian Research Council #177355/V30, and by the European Science Foundation Research Networking Programme HCAA.     

Universal methods of the search of normal geodesics on Lie groups with left-invariant sub-Riemannian metric

We obtain two versions of ODEs for the control function of normal geodesics for left-invariant sub-Riemannian metrics on Lie groups, involving only the structure of the Lie algebras of these groups. The first version is applicable to all Lie groups, while the second, to all matrix Lie groups; both versions are different invariant forms of the Hamiltonian system of the Pontryagin maximum principle for a left-invariant time-optimal problem on a Lie group. Basing on the first version, we find sufficient conditions for the normality of all geodesics of a given sub-Finslerian metric on a Lie group; in particular, we show that all three-dimensional Lie groups possess this property. The proofs use simple techniques of linear algebra.     

Sub-Riemannian calculus on hypersurfaces in Carnot groups

We develop a sub-Riemannian calculus for hypersurfaces in graded nilpotent Lie groups. We introduce an appropriate geometric framework, such as horizontal Levi-Civita connection, second fundamental form, and horizontal Laplace-Beltrami operator. We analyze the relevant minimal surfaces and prove some basic integration by parts formulas. Using the latter we establish general first and second variation formulas for the horizontal perimeter in the Heisenberg group. Such formulas play a fundamental role in the sub-Riemannian Bernstein problem.