A Coarea Formula for Smooth Contact Mappings of Carnot–Carathéodory Spaces

We prove the coarea formula for sufficiently smooth contact mappings of Carnot manifolds to Carnot–Carathéodory spaces. In particular, we investigate level surfaces of these mappings, and compare Riemannian and sub-Riemannian measures on them. Our main tool is the sharp asymptotic behavior of the Riemannian measure of the intersection of a tangent plane to a level surface and a sub-Riemannian ball. This calculation in particular implies that the sub-Riemannian measure of the set of characteristic points (i.e., the points at which the sub-Riemannian differential is degenerate) equals zero on almost every level set.     

Trace heat kernel asymptotics in 3D contact sub-Riemannian geometry

In this paper, we study the small time asymptotics for the heat kernel on a sub-Riemannian manifold, using a perturbative approach. We 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 terms of the two basic functional invariants χand κ defined on a 3D contact structure.     

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.     

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.     

On geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on distributions of corank 1

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 the Pontryagin maximum principle, we treat Riemannian and sub-Riemannian cases in a unified way and obtain some algebraic necessary conditions for the geodesic equivalence of (sub-)Riemannian metrics. In this way, first we obtain a new elementary proof of the classical Levi-Civita theorem on the classification of all Riemannian geodesically equivalent metrics in a neighborhood of the so-called regular (stable) point w.r.t. these metrics. Second, we prove that sub-Riemannian metrics on contact distributions are geodesically equivalent iff they are constantly proportional. Then we describe all geodesically equivalent sub-Riemannian metrics on quasi-contact distributions. Finally, we give a classification of all pairs of geodesically equivalent Riemannian metrics on a surface that are proportional at an isolated point. This is the simplest case, which was not covered by Levi-Civita’s theorem. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 21, Geometric Problems in Control Theory, 2004.     

A bilinear form associated to contact sub-conformal manifolds

We define a bilinear form associated to a sub-Riemannian contact manifold. It transforms by scalar multiples under sub-conformal transformations and with further hypothesis it is naturally defined on certain torus bundles over the contact manifold.     

On fat sub-Riemannian symmetric spaces in codimension three

A sub-Riemannian manifold is a differentiable manifold together with a smooth distribution which is equipped with a Riemannian metric. In this paper we attempt to study sub-Riemannian symmetric spaces (i.e., homogeneous sub-Riemannian manifolds admitting an involutive sub-Riemannian isometry at all points which is a central symmetry when restricted to the distribution) where the associated distribution is a codimension three fat distribution. We obtain a restricted classification theorem in dimension seven and we also construct a class of examples of quaternionic type in varying dimension.     

Invariant affinor and sub-Kähler structures on homogeneous spaces

We consider G-invariant affinor metric structures and their particular cases, sub-Kähler structures, on a homogeneous space G/H. The affinor metric structures generalize almost Kähler and almost contact metric structures to manifolds of arbitrary dimension. We consider invariant sub-Riemannian and sub-Kähler structures related to a fixed 1-form with a nontrivial radical. In addition to giving some results for homogeneous spaces of arbitrary dimension, we study these structures separately on the homogeneous spaces of dimension 4 and 5.     

The graphs of Lipschitz functions and minimal surfaces on Carnot groups

We study and solve a new problem for the class of Lipschitz mappings (with respect to sub-Riemannian metrics) on Carnot groups. We introduce the new concept of graph for the functions on a Carnot group, and then the new concept of sub-Riemannian differentiability generalizing hc-differentiability. We prove that the mapping-??graphs?? are almost everywhere differentiable in the new sense. For these mappings we define a concept of intrinsic measure and obtain an area formula for calculating this measure. By way of application, we find necessary and sufficient conditions on the class of surface-??graphs?? under which they are minimal surfaces (with respect to the intrinsic measure of a surface).     

A topological splitting theorem for sub-Riemannian manifolds

We prove an analogue of the Cheeger–Gromoll splitting theorem for sub-Riemannian manifolds with the measure contraction property instead of the nonnegativity of the Ricci curvature. If such a sub-Riemannian manifold contains a straight line, then the manifold splits diffeomorphically, where the splitting is not necessarily isometric. We prove that such a sub-Riemannian manifold containing a straight line cannot split isometrically under some typical condition in sub-Riemannian geometry. Heisenberg groups are such examples.     

On the Whitney extension property for continuously differentiable horizontal curves in sub-Riemannian manifolds

In this article we study the validity of the Whitney \(C^1\) extension property for horizontal curves in sub-Riemannian manifolds that satisfy a first-order Taylor expansion compatibility condition. We first consider the equiregular case, where we show that the extension property holds true whenever a suitable non-singularity property holds for the endpoint map on the Carnot groups obtained by nilpotent approximation. We then discuss the case of sub-Riemannian manifolds with singular points and we show that all step-2 manifolds satisfy the \(C^1\) extension property. We conclude by showing that the \(C^1\) extension property implies a Lusin-like approximation theorem for horizontal curves on sub-Riemannian manifolds.     

Quasiregular Mappings on Sub-Riemannian Manifolds

We study mappings on sub-Riemannian manifolds which are quasiregular with respect to the Carnot–Carathéodory 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 is not available in the setting of general sub-Riemannian manifolds. In the present paper we adopt therefore a metric rather than analytic viewpoint. As a first main result, we prove that the sub-Riemannian lens space admits nontrivial uniformly quasiregular (UQR) mappings, that is, quasiregular mappings with a uniform bound on the distortion of all the iterates. In doing so, we also obtain new examples of UQR maps on the standard sub-Riemannian spheres. The proof is based on a method for building conformal traps on sub-Riemannian spheres using quasiconformal flows, and an adaptation of this approach to quotients of spheres. One may then study the quasiregular semigroup generated by a UQR mapping. In the second part of the paper we follow Tukia to prove the existence of a measurable conformal structure which is invariant under such a semigroup. Here, the conformal structure is specified only on the horizontal distribution, and the pullback is defined using the Margulis–Mostow derivative (which generalizes the classical and Pansu derivatives).     

The polynomial sub-Riemannian differentiability of some Hölder mappings of Carnot groups

The polynomial sub-Riemannian differentiability is established for the large classes of Hölder mappings in the sub-Riemannian sense, namely, the classes of smooth mappings, their graphs, and the graphs of Lipschitz mappings in the sub-Riemannian sense defined on nilpotent graded groups. We also describe some special bases that carry the sub-Riemannian structure of the preimage to the image.     

Riemannian and Sub-Riemannian Geodesic Flows

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 result allows us to describe the sub-Riemannian geodesic flow on totally geodesic Riemannian foliations in terms of the Riemannian geodesic flow. Also, given a submersion \(\pi :M \rightarrow B\), we describe when the projections of a Riemannian and a sub-Riemannian geodesic flow in M coincide.     

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.     

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.     

Neurogeometry of neural functional architectures

The term “neurogeometry” denotes the geometry of the functional architecture of visual areas. The paper reviews some elements of the neurogeometry of the functional architecture of the first visual area V1 and explains why contact geometry, sub-Riemannian geometry, and noncommutative harmonic analysis are brought in as natural tools. It emphasizes the fact that these geometries are radically different from Riemannian geometries.     

Intrinsic complements of equiregular sub-Riemannian manifolds

Under a nondegeneracy condition, we show that an equiregular sub-Riemannian manifold of step size \(r\) admits a canonical, \(V\) -rigid complement defined from the sub-Riemannian data that is preserved the by action of sub-Riemannian isometries. We explore how the existence of such a complement relates to results from the literature and study the step size 2 case in more detail.     

On a Step 2(k + 1) sub-Riemannian manifold

We compute the sub-Riemannian distance for a Step 2(k + 1) sub-Riemannian manifold from the origin to any given point. We characterize the number of sub-Riemannian geodesics between the origin and any other point.     

Any sub-Riemannian metric has points of smoothness

We prove the result stated in the title that is equivalent to the existence of a regular point of the sub-Riemannian exponential mapping. In the case of a complete real-analytic sub-Riemannian manifold, we prove that the metric is analytic on an open everywhere dense subset. Published in Doklady Akademii Nauk, 2009, Vol. 424, No. 3, pp. 295–298. Presented by Academician E.F. Mishchenko September 1, 2008 The article was translated by the author.