On Local Approximation Theorem on Equiregular Carnot-Carathéodory Spaces

Карманова, М. Б.; Vodop’yanov, S. K.

doi:10.1007/978-3-319-02132-4_15

Cited by 10 publications

(9 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…It was formulated in [G, p. 135] for "sufficiently smooth vector fields". It was proved in [VK2] for C 1,α -smooth vector fields but the same arguments work for the case of C 1 -smooth vector fields since they are based on the property (1.3) [KV1,Theorem 7].…”

Section: Carnot-carathéodory Metric and Ball-box Theoremmentioning

confidence: 87%

See 1 more Smart Citation

Approximate Differentiability of Mappings of Carnot-Carathéodory Spaces

Basalaev,

Vodopyanov

2012

Preprint

Self Cite

View full text Add to dashboard Cite

We study the approximate differentiability of measurable mappings of Carnot-Carathéodory spaces. We show that the approximate differentiability almost everywhere is equivalent to the approximate differentiability along the basic horizontal vector fields almost everywhere. As a geometric tool we prove the generalization of Rashevsky-Chow theorem for C 1 -smooth vector fields. The main result of the paper extends theorems on approximate differentiability proved by Stepanoff (1923Stepanoff ( , 1925 and Whitney (1951) in Euclidean spaces and by Vodopyanov (2000) on Carnot groups. Contents

show abstract

Section: Carnot-carathéodory Metric and Ball-box Theoremmentioning

confidence: 87%

“…Observe, that the latter implies d g ∞ (g, x) = d ∞ (g, x). Proposition 1.6 ( [KV,KV1]). The quantity d ∞ is a quasimetric in the sense of [NSW] that is the following relations hold for all points of the neighborhood…”

Section: Local Geometry Of Carnot-carathéodory Spacesmentioning

confidence: 99%

Approximate Differentiability of Mappings of Carnot-Carathéodory Spaces

Basalaev,

Vodopyanov

2012

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Пусть x, y ∈ M. Расстоянием Карно-Каратеодори d cc (x, y) называется величина [16], [19]), но более удобную для работы.…”

unclassified

“…Несмотря на то, что квазиметрика в теореме 3.4 отличается от квазиметрики, изученной в [16], [18] и [19], утверждение верно, так как схема доказательства для d 2 та же самая, что и для d ∞ .…”

unclassified

See 1 more Smart Citation

Формула Площади Для Липшицевых Отображений Пространств Карно - Каратеодори

Карманова¹,

Карманова²

2014

Известия Российской академии наук. Серия математическая

View full text Add to dashboard Cite

Metric Geometry of Nonregular Weighted Carnot–Carathéodory Spaces

Selivanova

2013

J Dyn Control Syst

View full text Add to dashboard Cite

We investigate local and metric geometry of weighted Carnot-Carathéodory spaces which are a wide generalization of sub-Riemannian manifolds and arise in nonlinear control theory, subelliptic equations etc. For such spaces the intrinsic Carnot-Carathéodory metric might not exist, and some other new effects take place. We describe the local algebraic structure of such a space, endowed with a certain quasimetric (first introduced by A. Nagel, E.M. Stein and S. Wainger), and compare local geometries of the initial C-C space and its tangent cone at some fixed (possibly nonregular) point. The main results of the present paper are new even for the case of sub-Riemannian manifolds. Moreover, they yield new proofs of such classical results as the Local approximation theorem and the Tangent cone theorem, proved for Hörmander vector fields by M. Gromov, A.Bellaiche, J.Mitchell etc.

show abstract

On Local Approximation Theorem on Equiregular Carnot-Carathéodory Spaces

Cited by 10 publications

References 26 publications

Approximate Differentiability of Mappings of Carnot-Carathéodory Spaces

Approximate Differentiability of Mappings of Carnot-Carathéodory Spaces

Формула Площади Для Липшицевых Отображений Пространств Карно - Каратеодори

Metric Geometry of Nonregular Weighted Carnot–Carathéodory Spaces

Contact Info

Product

Resources

About