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Sibirsk. Mat. Zh., 2007, Volume 48, Number 2, Pages 251–271 (Mi smj22)  

This article is cited in 18 scientific papers (total in 18 papers)

Differentiability of mappings in the geometry of Carnot manifolds

S. K. Vodop'yanov

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: We study the differentiability of mappings in the geometry of Carnot–Carathéodory spaces under the condition of minimal smoothness of vector fields. We introduce a new concept of $hc$-differentiability and prove the $hc$-differentiability of Lipschitz mappings of Carnot–Carathéodory spaces (a generalization of Rademacher's theorem) and a generalization of Stepanov's theorem. As a consequence, we obtain the $hc$-differentiability almost everywhere of the quasiconformal mappings of Carnot–Carathéodory spaces. We establish the $hc$-differentiability of rectifiable curves by way of proof. Moreover, the paper contains a new proof of the functorial property of the correspondence “a local basis $\mapsto$ the nilpotent tangent cone.”

Keywords: Carnot–Carathéodory space, subriemannian geometry, nilpotent tangent cone, differentiability of curves and Lipschitz mappings.

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English version:
Siberian Mathematical Journal, 2007, 48:2, 197–213

Bibliographic databases:

UDC: 514.763.22+517.518.15+514.752.8
Received: 28.06.2004
Revised: 12.02.2007

Citation: S. K. Vodop'yanov, “Differentiability of mappings in the geometry of Carnot manifolds”, Sibirsk. Mat. Zh., 48:2 (2007), 251–271; Siberian Math. J., 48:2 (2007), 197–213

Citation in format AMSBIB
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\yr 2007
\vol 48
\issue 2
\pages 251--271
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\jour Siberian Math. J.
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\issue 2
\pages 197--213
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. S. K. Vodop'yanov, D. V. Isangulova, “Differentiability of the mappings of Carnot–Caratheodory spaces in the Sobolev and $BV$-topologies”, Siberian Math. J., 48:1 (2007), 37–55  mathnet  crossref  mathscinet  zmath  isi  elib
    2. Vodopyanov S.K., “Geometry of Carnot-Carathéodory spaces and differentiability of mappings”, The interaction of analysis and geometry, Contemp. Math., 424, 2007, 247–301  crossref  mathscinet  zmath  isi
    3. Vodop'yanov S.K., Karmanova M.B., “Sub-Riemannian Geometry for Vector Fields of Minimal Smoothness”, Dokl. Math., 78:2 (2008), 737–742  mathnet  crossref  mathscinet  zmath  isi  elib  scopus
    4. A. V. Greshnov, “On applications of the Taylor formula in some quasispaces”, Siberian Adv. Math., 20:3 (2010), 164–179  mathnet  crossref  mathscinet  elib  elib
    5. Vodop'yanov S.K., Selivanova S.V., “Algebraic properties of the tangent cone to a quasimetric space with dilations”, Dokl. Math., 80:2 (2009), 734–738  mathnet  crossref  mathscinet  zmath  isi  elib  scopus
    6. Selivanova S.V., “Tangent cone to a regular quasimetric Carnot–Carathéodory space”, Dokl. Math., 79:2 (2009), 265–269  mathnet  crossref  mathscinet  zmath  isi  elib  elib  scopus
    7. Karmanova M., Vodop'yanov S., “Geometry of Carnot-Carathéodory spaces, differentiability, coarea and area formulas”, Analysis and mathematical physics, Trends Math., Birkhäuser, Basel, 2009, 233–335  mathscinet  zmath  isi  elib
    8. S. V. Selivanova, “The tangent cone to a quasimetric space with dilations”, Siberian Math. J., 51:2 (2010), 313–324  mathnet  crossref  mathscinet  isi
    9. Buliga M., “Infinitesimal affine geometry of metric spaces endowed with a dilatation structure”, Houston Journal of Mathematics, 36:1 (2010), 91–136  mathscinet  zmath  isi
    10. A. V. Greshnov, “On one class of Lipschitz vector fields in $\mathbb R^3$”, Siberian Math. J., 51:3 (2010), 410–418  mathnet  crossref  mathscinet  zmath  isi
    11. A. D. Kozhevnikov, “Inverse and implicit function theorems on Carnot manifolds”, Siberian Math. J., 51:6 (2010), 1047–1060  mathnet  crossref  mathscinet  isi
    12. Karmanova M.B., “Convergence of scaled vector fields and local approximation theorem on Carnot-Carathéodory spaces and applications”, Dokl. Math., 84:2 (2011), 711–717  crossref  mathscinet  mathscinet  zmath  zmath  isi  elib  elib  scopus
    13. Selivanova S.V. Vodopyanov S.K., “Algebraic and Analytic Properties of Quasimetric Spaces with Dilations”, Complex Analysis and Dynamical Systems IV, Pt 1: Function Theory and Optimization, Contemporary Mathematics, 553, ed. Agranovsky M. BenArtzi M. Galloway G. Karp L. Reich S. Shoikhet D. Weinstein G. Zalcman L., Amer Mathematical Soc, 2011, 267–287  crossref  mathscinet  zmath  isi
    14. Q. Y. Wu, W. Wang, “The Beltrami equations for quasiconformal mappings on strongly pseudoconvex hypersurfaces”, Siberian Math. J., 53:2 (2012), 316–334  mathnet  crossref  mathscinet  isi
    15. S. G. Basalaev, S. K. Vodopyanov, “Approximate differentiability of mappings of Carnot–Carathéodory spaces”, Eurasian Math. J., 4:2 (2013), 10–48  mathnet  mathscinet  zmath
    16. Bigolin F., Kozhevnikov A., “Tangency, Paratangency and Four-Cones Coincidence Theorem in Carnot Groups”, J. Convex Anal., 21:3 (2014), 887–899  mathscinet  zmath  isi
    17. Karmanova M. Vodopyanov S., “on Local Approximation Theorem on Equiregular Carnot-Caratheodory Spaces”, Geometric Control Theory and Sub-Riemannian Geometry, Springer Indam Series, 4, ed. Stefani G. Boscain U. Gauthier J. Sarychev A. Sigalotti M., Springer Int Publishing Ag, 2014, 241–262  crossref  mathscinet  zmath  isi  scopus
    18. M. V. Tryamkin, “The morphism property of subelliptic equations on the roto-translation group”, Siberian Math. J., 56:5 (2015), 936–954  mathnet  crossref  crossref  isi  elib  elib
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