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

  • Conference paper
Analysis and Mathematical Physics

Part of the book series: Trends in Mathematics ((TM))

Abstract

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 C1,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.

The research was partially supported by the Commission of the European Communities (Specific Targeted Project “Geometrical Analysis in Lie groups and Applications”, Contract number 028766), the Russian Foundation for Basic Research (Grant 08-01-00531), the State Maintenance Program for Young Russian Scientists and the Leading Scientific Schools of Russian Federation (Grant NSh-5682.2008.1).

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Pr-t Akademika Koptyuga 4, Novosibirsk, 630090, Russia

    Maria Karmanova & Sergey Vodop′yanov

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  1. Maria Karmanova
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  2. Sergey Vodop′yanov
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  1. Department of Mathematics, Royal Institute of Technology (KTH), 100 44, Stockholm, Sweden

    Björn Gustafsson

  2. Department of Mathematics, University of Bergen, Johannes Brunsgate 12, 5008, Bergen, Norway

    Alexander Vasil’ev

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Karmanova, M., Vodop′yanov, S. (2009). Geometry of Carnot-Carathéodory Spaces, Differentiability, Coarea and Area Formulas. In: Gustafsson, B., Vasil’ev, A. (eds) Analysis and Mathematical Physics. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-9906-1_14

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