Metric Geometry of Nonregular Weighted Carnot–Carathéodory Spaces (original) (raw)

Geometry of Carnot-Carathéodory Spaces, Differentiability, Coarea and Area Formulas

Analysis and Mathematical Physics, 2009

We compare geometries of two different local Lie groups in a Carnot-Carathéodory space, and obtain quantitative estimates of their difference. This result is extended to Carnot-Carathéodory spaces with C 1,α -smooth basis vector fields, α ∈ [0, 1], and the dependence of the estimates on α is established. From here we obtain the similar estimates for comparing geometries of a Carnot-Carathéodory space and a local Lie group. These results base on Gromov's Theorem on nilpotentization of vector fields for which we give new and simple proof. All the above imply basic results of the theory: Gromov type Local Approximation Theorems, and for α > 0 Rashevskiǐ-Chow Theorem and Ball-Box Theorem, etc. We apply the obtained results for proving hc-differentiability of mappings of Carnot-Carathéodory spaces with continuous horizontal derivatives. The latter is used in proving the coarea formula for smooth contact mappings of Carnot-Carathéodory spaces, and the area formula for Lipschitz (with respect to sub-Riemannian metrics) mappings of Carnot-Carathéodory spaces.

Geometry of Carnot-Carath'eodory Spaces, Differentiability and Coarea Formula

2009

We give a simple proof of Gromov's Theorem on nilpotentization of vector fields, and exhibit a new method for obtaining quantitative estimates of comparing geometries of two different local Carnot groups in Carnot-Carathéodory spaces with C 1,α-smooth basis vector fields, α ∈ [0, 1]. From here we obtain the similar estimates for comparing geometries of a Carnot-Carathéodory space and a local Carnot group. These two theorems imply basic results of the theory: Gromov type Local Approximation Theorems, and for α > 0 Rashevskiǐ-Chow Theorem and Ball-Box Theorem, etc. We apply the obtained results for proving hc-differentiability of mappings of Carnot-Carathéodory spaces with continuous horizontal derivatives. The latter is used in proving the coarea formula for some classes of contact mappings of Carnot-Carathéodory spaces.

Algebraic properties of the tangent cone to a quasimetric space with dilations

Doklady Mathematics, 2009

In this paper we study algebraic and analytic prop erties of topological spaces on which dilations (that is, one parameter families of contractive homeomor phisms acting in a neighborhood of each point) are defined and, in particular, properties of the tangent cone to a (quasi)metric space endowed with a dilation structure. A motivation for studying spaces with dila tions can be found in, e.g., . The system of axi oms introduced below ensures the existence of a tan gent cone at any fixed point of a quasimetric space. The notion of the tangent cone to a metric space intro duced by Gromov (see ) generalizes that of the tan gent space to a smooth manifold and is useful in study ing many questions of the theory of metric spaces, in particular, Carnot-Carathéodory spaces (see [1, 2, 4-7, 10, 11] and the references therein). A motivation for studying quasimetric spaces (see Definition 1) is given in, e.g., . The generalization of Gromov's theory to quasimetric spaces meets some obstacles, which were overcome in . In [10], the existence of a tangent cone to a regular quasimetric Carnot-Car athéodory space was also proved. Using the method of [10], we prove the existence of a tangent cone to a space with dilations (see Definition 2, axioms (A0)-(A3)). Note that these axioms alone imply no addi tional properties of the tangent cone.

Algebraic and analytic properties of quasimetric spaces with dilations

Contemporary Mathematics, 2011

We provide an axiomatic approach to the theory of local tangent cones of regular sub-Riemannian manifolds and the differentiability of mappings between such spaces. This axiomatic approach relies on a notion of a dilation structure which is introduced in the general framework of quasimetric spaces. Considering quasimetrics allows us to cover a general case including, in particular, minimal smoothness assumptions on the vector fields defining the sub-Riemannian structure. It is important to note that the theory existing for metric spaces can not be directly extended to quasimetric spaces.

SOME RESULTS ON SUB-RIEMANNIAN GEOMETRY

Sub-Riemannian structures naturally occur in different branches of Mathematics in the study of constrained systems in classical mechanics, in optimal control, geometric measure theory and differential geometry. In this paper, we show that Sub-Riemannian structures on three manifolds locally depend on two functions 1 and K of three variables and we investigate how these differential invariants influence the geometry.

Variational problems concerning sub-Finsler metrics in Carnot groups

2022

This paper is devoted to the study of geodesic distances defined on a subdomain of a given Carnot group, which are bounded both from above and from below by fixed multiples of the Carnot–Carathéodory distance. We show that the uniform convergence (on compact sets) of these distances can be equivalently characterized in terms of Γ-convergence of several kinds of variational problems. Moreover, we investigate the relation between the class of intrinsic distances, their metric derivatives and the sub-Finsler convex metrics defined on the horizontal bundle.