Subanalyticity of the Sub-Riemannian Distance

The aim of this paper is to study the subanalyticity of the distance function d defined by a sub-Riemannian structure (, g). If the distribution is of degree 2, we prove that d is subanalytic and if ² is fat d is subanalytic far away from the diagonal. In this last case we prove in fact that the function d(x ₀,·) is subanalytic in a neighborhood of x ₀.     

k-Step Sub-Riemannian Manifold whose Sub-Riemannian Metric Admits a Canonical Extension to a Riemannian Metric

We show that the metric of a k-step bracket generating sub-Riemannian manifold can be canonically extended to a Riemannian metric, provided the distribution D is an exporter. This is a generic condition. For such manifolds, we construct a canonical connection ? with nonzero torsion, adapted to the filtration D = D ¹ ? D ²…? D ^k = TM. In some cases, the geodesics related to this adapted connection and tangent to D are normal Carnot–Caratheodory geodesics.     

Examples of Integrable Sub-Riemannian Geodesic Flows

We exhibit examples of sub-Riemannian metrics with integrable geodesic flows and positive topological entropy. The underlying distribution is contact in 3 dimensions and is Goursat in the higher-dimensional case. We also give an example of Poisson ³-action on 6-dimensional manifold such that topological entropies for all nonzero vectors (Hamiltonians) are positive.     

On Three Problems About Dynamical Systems Related to Lattices and Homogeneous Spaces

This communication is a survey about the problems discussed recently in my seminar at Moscow State University. Precisely the topics are as follows: (1) the Veech dichotomy for rational plane billiards, (2) T-induced flows, and (3) integrability of G-invariant Hamiltonian systems with homogeneous configuration spaces.     

Sub-Riemannian structures on 3D lie groups

We give a complete classification of left-invariant sub-Riemannian structures on three-dimensional Lie groups in terms of the basic differential invariants. As a consequence, we explicitly find a sub-Riemannian isometry between the nonisomorphic Lie groups SL(2) and A ⁺( mathbbR mathbb{R} ) × S ¹, where A ⁺( mathbbR mathbb{R} ) denotes the group of orientation preserving affine maps on the real line.     

Sub-Riemannian Curvature of Carnot Groups with Rank-Two Distributions

The notion of curvature discussed in this paper is a far-going generalization of the Riemannian sectional curvature. It was first introduced by Agrachev et al. ([2015]), and it is defined for a wide class of optimal control problems: a unified framework including geometric structures such as Riemannian, sub-Riemannian, Finsler, and sub-Finsler structures. In this work, we study the generalized sectional curvature of Carnot groups with rank-two distributions. In particular, we consider the Cartan group and Carnot groups with horizontal distribution of Goursat-type. In these Carnot groups, we characterize ample and equiregular geodesics. For Carnot groups with horizontal Goursat distribution, we show that their generalized sectional curvatures depend only on the Engel part of the distribution. This family of Carnot groups contains naturally the three-dimensional Heisenberg group, as well as the Engel group. Moreover, we also show that in the Engel and Cartan groups, there exist initial covectors for which there is an infinite discrete set of times at which the corresponding ample geodesics are not equiregular.     

Nilpotent (3, 6) Sub-Riemannian Problem

In this paper we study the nilpotent (3, 6) sub-Riemannian problem. We describe the envelope of sub-Riemannian geodesics starting from a fixed point. We also describe the wave fronts propagating from the point. For general nilpotent (n,n(n + 1)/2) sub-Riemannian problem we formulate a conjecture about the form of the variety where geodesics starting from a fixed point lose optimality.     

A Nonintegrable Sub-Riemannian Geodesic Flow on a Carnot Group

Graded nilpotent Lie groups, orCarnot groups, are to sub-Riemannian geometry as Euclidean spaces are to Riemannian geometry. They are the metric tangent cones for this geometry. Hoping that the analogy between sub-Riemannian and Riemannian geometry is a strong one, one might conjecture that the sub-Riemannian geodesic flow on any Carnot group is completely integrable. We prove this conjecture to be false by showing that the sub-Riemannian geodesic flow is not algebraically completely integrable in the case of the group whose Lie algebra consists of 4 by 4 upper triangular matrices. As a corollary, we prove that the centralizer for the corresponding quadratic “quantum” Hamiltonian in the universal enveloping algebra of this Lie algebra is “as small as possible.”     

On Sub-Riemannian and Riemannian Structures on the Heisenberg Groups

We consider the left-invariant sub-Riemannian and Riemannian structures on the Heisenberg groups. A classification of these structures was found previously. In the present paper, we find (for each normalized structure) the isometry group, the exponential map, the totally geodesic subgroups, and the conjugate locus. Finally, we determine the minimizing geodesics from identity to any given endpoint. (Several of these points have been covered, to varying degrees, by other authors.)     

Sub-Riemannian Geometry: One-Parameter Deformation of the Martinet Flat Case

Consider a sub-Riemannian geometry (U,D,g), where U is a neighborhood of 0 in ³, D is a Martinet type distribution identified to ker , being the one-form and g is a metric on D which can be written as a(q)dx ² + 2b(q)dxdy + c(q)dy ², whereq = (x,y,z). In a previous article [1] we proved that g can be written in a normal form where b 0, a = 1 + yF(q ), c = 1 + G(q ), where . Moreover we analyzed the flat case a = c = 1. In this article we study the following one-parameter deformation of the flat case: a = l, c = (1 + y)² where µ . We parametrize the set of geodesics using elliptic functions. This allows us to compute the trace of the sphere and the wave front of small radius on the plane y = 0. We show that the sphere of small radius is not sub-analytic. This analysis clarifies the role of one of the functional invariants in the normal form.     

Conjugate Time in the Sub-Riemannian Problem on the Cartan Group

Journal of Dynamical and Control Systems - The Cartan group is the free nilpotent Lie group of rank 2 and step 3. We consider the left-invariant sub-Riemannian problem on the Cartan group defined...     

Top Limit Characteristics of New Type Cascade PC on the Basis of Homogeneous Semiconductor

A new type homogeneous planar PC （photoelectric converter） on the basis of multijunction semiconductor n＋-p-p＋-n＋-p-p＋-...-n＋-p-p＋ structure has been investigated. The entire structure is a cascade PC consisting of a number of elements of the structure--single PCs connected in series and illuminated by light that has consistently passed through the previous semiconductor layers. The theory of converter of both monochromatic and solar radiation has been developed and the limiting values of their photoelectric and power characteristics have been determined, including the optimal thickness and number of single PCs layered on a base PC, their spectral sensitivity, current-voltage characteristics and efficiency. The open-circuit voltage grows practically linearly with the number of elements in the cascade. The top efficiency limit for a certain optimal elements number reaches its maximum that exceeds considerably that of the base PC, especially in the range of low collecting coefficient of charge carriers in the base PC.     

On Sub-Riemannian Caustics and Wave Fronts for Contact Distributions in the Three-Space

In a number of previous papers of the first and third authors, caustics, cut-loci, spheres, and wave fronts of a system of sub-Riemannian geodesics emanating from a point q ₀ were studied. It turns out that only certain special arrangements of classical Lagrangian and Legendrian singularities occur outside q ₀. As a consequence of this, for instance, the generic caustic is a globally stable object outside the origin q ₀. Here we solve two remaining stability problems. The first part of the paper shows that in fact generic caustics have moduli at the origin, and the first module that occurs has a simple geometric interpretation. On the contrary, the second part of the paper shows a stability result at q ₀. We define the big wave front: it is the graph of the multivalued function arclength wave-front reparametrized in a certain way. This object is a three-dimensional surface that also has a natural structure of the wave front. The projection of the singular set of this big wave front on the 3-dimensional space is nothing else but the caustic. We show that in fact this big wave front is Legendre-stable at the origin.     

Geodesics in the Engel Group with a Sub-Lorentzian Metric

Let E be the Engel group and D be a rank 2 bracket generating left invariant distribution with a Lorentzian metric, which is a nondegenerate metric of index 1. In this paper, we first study some properties of horizontal curves on E. Second, we prove that time-like normal geodesics are locally maximizers in the Engel group and calculate the explicit expression of non-space-like geodesics.     

First Variation of the Hausdorff Measure of Non-horizontal Submanifolds in Sub-Riemannian Stratified Lie Groups

We determine necessary conditions for a non-horizontal submanifold of a sub-Riemannian stratified Lie group to be of minimal measure. We calculate the first variation of the measure for a non-horizontal submanifold and find that the minimality condition implies the tensor equation H + σ = 0, where H is analogous to the mean curvature and σ is the mean torsion. We also discuss new examples of minimal non-horizontal submanifolds in the Heisenberg group, in particular surfaces in \(\mathbb {H}^{2}\).     

Correction to: Conjugate Time in the Sub-Riemannian Problem on the Cartan Group

Journal of Dynamical and Control Systems - A correction to this paper has been published: https://doi.org/10.1007/s10883-021-09569-8 .     

Extremal Trajectories and Maxwell Strata in Sub-Riemannian Problem on Group of Motions of Pseudo-Euclidean Plane

We consider the sub-Riemannian length minimization problem on the group of motions of pseudo-Euclidean plane that form the special hyperbolic group SH(2). The system comprises of left invariant vector fields with 2-dimensional linear control input and energy cost functional. We apply the Pontryagin maximum principle to obtain the extremal control input and the sub-Riemannian geodesics. A change of coordinates transforms the vertical subsystem of the normal Hamiltonian system into the mathematical pendulum. In suitable elliptic coordinates, the vertical and the horizontal subsystems are integrated such that the resulting extremal trajectories are parametrized by the Jacobi elliptic functions. Qualitative analysis reveals that the projections of normal extremal trajectories on the xy-plane have cusps and inflection points. The vertical subsystem being a generalized pendulum admits reflection symmetries that are used to obtain a characterization of the Maxwell strata.     

Transformations Having Homogeneous Spectra

We show that for a generic automorphism T, the Cartesian product T × T has homogeneous spectrum of multiplicity two. New examples of automorphisms with the property * are presented.     

Regions Where the Exponential Map at Regular Points of Sub-Riemannian Manifolds is a Local Diffeomorphism

For a k-step sub-Riemannian manifold which admits a bracket generating vector at a point, we describe a region near the point where the exponential map is a local diffeomorphism. This is proved by taking the Taylor series of the exponential map and calculating the first nonzero term, which has order , where n is the topological dimension and is the Hausdorff dimension of the metric space associated to the sub-Riemannian manifold.