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Differentiability of mappings in the geometry of Carnot manifolds

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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 ↦ the nilpotent tangent cone.”

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References

  1. Goodman R. W., Nilpotent Lie Groups: Structure and Applications to Analysis, Springer-Verlag, Berlin (1970) (Lecture Notes in Math.; Vol. 562).

    Google Scholar 

  2. Rothschild L. and Stein E., “Hypoelliptic differential operators and nilpotent groups,” Acta Math., 137, 247–320 (1976).

    Article  MathSciNet  Google Scholar 

  3. Metivier G., “Fonction spectrale et valeurs proposes d’une classe d’operateurs,” Comm. Partial Differential Equations, 1, 479–519 (1976).

    MathSciNet  Google Scholar 

  4. Nagel A., Stein E. M., and Wainger S., “Balls and metrics defined by vector fields. I: Basic properties,” Acta Math., 155, 103–147 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  5. Vershik A. M. and Gershkovich V. Ya., “Nonholonomic dynamical systems, geometry of distributions and variational problems,” in: Dynamical Systems. VII. Encycl. Math. Sci. Vol. 16, 1994, pp. 1–81.

  6. Gromov M., “Carnot-Carathéodory spaces seen from within,” in: Sub-Riemannian Geometry, Birkhäuser, Basel, 1996, pp. 79–323.

    Google Scholar 

  7. Pansu P., “Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un,” Ann. Math. (2), 129, No. 1, 1–60 (1989).

    Article  MathSciNet  Google Scholar 

  8. Vodopyanov S. K., “-Differentiability on Carnot groups in different topologies and related topics,” in: Studies on Analysis and Geometry, Sobolev Institute, Novosibirsk, 2000, pp. 603–670.

    Google Scholar 

  9. Vodopyanov S. K. and Ukhlov A. D., “Approximately differentiable transformations and change of variables on nilpotent groups,” Siberian Math. J., 37, No. 1, 62–78 (1996).

    Article  MathSciNet  Google Scholar 

  10. Magnani V., “Differentiability and area formulas on stratified Lie groups,” Houston J. Math., 27, No. 2, 297–323 (2001).

    MATH  MathSciNet  Google Scholar 

  11. Rademacher H., “Ueber partielle und totale Differenzierbarkeit. I,” Math. Ann., Bd 79, 340–359 (1919).

    Article  MathSciNet  Google Scholar 

  12. Stepanoff W., “Ueber totale Differenzierbarkeit,” Math. Ann., Bd 90, 318–320 (1923).

    Article  MathSciNet  MATH  Google Scholar 

  13. Folland G. B. and Stein E. M., Hardy Spaces on Homogeneous Groups, Princeton Univ. Press, Princeton (1982) (Math. Notes, Vol. 28).

    MATH  Google Scholar 

  14. Vodopyanov S. K. and Karmanova M. B., “Local geometry of Carnot manifolds under minimal smoothness,” Dokl. Akad. Nauk, 413, No. 3, 305–311 (2007).

    MathSciNet  Google Scholar 

  15. Vodopyanov S. K. and Karmanova M. B., “Subriemannian geometry under minimal smoothness of vector fields,” Dokl. Akad. Nauk (2007) (to appear).

  16. Karmanova M. and Vodopyanov S., Geometry of Carnot Manifolds Under Minimal Smoothness [Preprint, No. 183], Sobolev Institute, Novosibirsk (2006).

    Google Scholar 

  17. Vodopyanov S. K., “Differentiability of curves in the category of Carnot manifolds,” Doklady Math., 74, No. 2, 686–691 (2006).

    Article  Google Scholar 

  18. Vodopyanov S. K., “Differentiability of mappings of Carnot manifolds and an isomorphism of tangent cones,” Dokl. Akad. Nauk, 411, No. 4, 439–443 (2006).

    MathSciNet  Google Scholar 

  19. Vodopyanov S. K. and Greshnov A. V., “On the differentiability of mappings of Carnot-Carathéodory spaces,” Doklady Math., 67, No. 2, 246–251 (2003).

    Google Scholar 

  20. Vodopyanov S. K., “Geometry of Carnot-Carathéodory spaces and differentiability of mappings,” in: The Interaction of Analysis and Geometry, Amer. Math. Soc., Providence, RI, 2007, pp. 247–302 (Contemp. Math.; Vol. 424).

    Google Scholar 

  21. Vodopyanov S., Differentiability of Sobolev Mappings of Carnot-Carathéodory Spaces [Preprint, No. 182], Sobolev Institute, Novosibirsk (2006).

    Google Scholar 

  22. Hörmander L., “Hypoelliptic second order differential equations,” Acta Math., 119, 147–171 (1967).

    Article  MATH  MathSciNet  Google Scholar 

  23. Helgason S., Differential Geometry and Symmetric Spaces, Academic Press, New York; London (1962) (Pure Appl. Math.; Vol. 12).

    MATH  Google Scholar 

  24. Belläiche A., “The tangent space in sub-Riemannian geometry,” in: Sub-Riemannian Geometry, Birkhäuser, Basel, 1996, pp. 1–78.

    Google Scholar 

  25. Montgomery R., A Tour of Subriemannian Geometries, Their Geodesics and Applications, Amer. Math. Soc., Providence, RI (2002).

    MATH  Google Scholar 

  26. Pontryagin L. S., Ordinary Differential Equations, Addison-Wesley Publishing Company, Reading, Mass. etc.; Pergamon Press, London; Paris (1962) (Adiwes Internat. Ser. in Math.).

    MATH  Google Scholar 

  27. Vodopyanov S. K. and Ukhlov A. D., “Set functions and their applications in the theory of Lebesgue and Sobolev spaces. I,” Siberian Adv. Math., 14, No. 4, 78–125 (2004).

    MathSciNet  Google Scholar 

  28. Burago D., Burago Yu., and Ivanov S., A Course in Metric Geometry, Amer. Math. Soc., Providence, RI (2001).

    MATH  Google Scholar 

  29. Margulis G. A. and Mostow G. D., “The differential of a quasi-conformal mapping of a Carnot-Carathéodory space,” Geom. Funct. Anal., 5, No. 2, 402–433 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  30. Mitchell J., “On Carnot-Carathéodory metrics,” J. Differential Geometry, 21, 35–45 (1985).

    MATH  Google Scholar 

  31. Vodopyanov S. K., “Sobolev classes and quasiconformal mappings on Carnot-Carathéodory spaces,” in: Geometry, Topology and Physics. Proc. of the First Brazil-USA Workshop held in Campinas, Brazil, June 30–July 7, 1996. Eds.: B. N. Apanasov, S. B. Bradlow, W. A. Rodrigues, Jr., K. K. Uhlenbeck, Walter de Gruyter & Co, Berlin; New York, 1997, pp. 301–316.

    Google Scholar 

  32. Margulis G. A. and Mostow G. D., “Some remarks on the definition of tangent cones in a Carnot-Carathéodory space,” J. Anal. Math., 80, 299–317 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  33. Agrachev A. and Marigo A., “Nonholonomic tangent spaces: intrinsic construction and rigid dimensions,” Electron. Res. Announc. Amer. Math. Soc., 9, 111–120 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  34. Greshnov A. V., “Metrics and tangent cones of uniformly regular Carnot-Carathéodory spaces,” Siberian Math. J., 47, No. 2, 209–238 (2006).

    Article  MathSciNet  Google Scholar 

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  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    S. K. Vodopyanov

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Original Russian Text Copyright © 2007 Vodopyanov S. K.

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 48, No. 2, pp. 251–271, March–April, 2007.

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Vodopyanov, S.K. Differentiability of mappings in the geometry of Carnot manifolds. Sib Math J 48, 197–213 (2007). https://doi.org/10.1007/s11202-007-0022-4

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  • DOI: https://doi.org/10.1007/s11202-007-0022-4

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