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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
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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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This publication is cited in the following articles:
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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
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Vodopyanov S.K., “Geometry of Carnot-Carathéodory spaces and differentiability of mappings”, The interaction of analysis and geometry, Contemp. Math., 424, 2007, 247–301
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Vodop'yanov S.K., Karmanova M.B., “Sub-Riemannian Geometry for Vector Fields of Minimal Smoothness”, Dokl. Math., 78:2 (2008), 737–742
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A. V. Greshnov, “On applications of the Taylor formula in some quasispaces”, Siberian Adv. Math., 20:3 (2010), 164–179
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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
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Selivanova S.V., “Tangent cone to a regular quasimetric Carnot–Carathéodory space”, Dokl. Math., 79:2 (2009), 265–269
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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
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S. V. Selivanova, “The tangent cone to a quasimetric space with dilations”, Siberian Math. J., 51:2 (2010), 313–324
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Buliga M., “Infinitesimal affine geometry of metric spaces endowed with a dilatation structure”, Houston Journal of Mathematics, 36:1 (2010), 91–136
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A. V. Greshnov, “On one class of Lipschitz vector fields in $\mathbb R^3$”, Siberian Math. J., 51:3 (2010), 410–418
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A. D. Kozhevnikov, “Inverse and implicit function theorems on Carnot manifolds”, Siberian Math. J., 51:6 (2010), 1047–1060
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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
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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
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Q. Y. Wu, W. Wang, “The Beltrami equations for quasiconformal mappings on strongly pseudoconvex hypersurfaces”, Siberian Math. J., 53:2 (2012), 316–334
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S. G. Basalaev, S. K. Vodopyanov, “Approximate differentiability of mappings of Carnot–Carathéodory spaces”, Eurasian Math. J., 4:2 (2013), 10–48
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Bigolin F., Kozhevnikov A., “Tangency, Paratangency and Four-Cones Coincidence Theorem in Carnot Groups”, J. Convex Anal., 21:3 (2014), 887–899
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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
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M. V. Tryamkin, “The morphism property of subelliptic equations on the roto-translation group”, Siberian Math. J., 56:5 (2015), 936–954
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