Skip to main content
SpringerLink
Log in

On the generalized triangle inequality for quasimetrics induced by noncommuting vector fields

  • Published:
Siberian Advances in Mathematics Aims and scope Submit manuscript

Abstract

For a sufficiently wide class of r-smooth basis vector fields, we obtain necessary and sufficient conditions for some anisotropic metric functions induced by these vector fields to be quasimetrics. These results are applied to the problem of the existence of a nilpotent tangent cone at a distinguished point.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. A. Akivis, “The canonical expansions of the equations of a local analytic quasigroup”, Dokl. Akad. Nauk SSSR 188(5), 967–970 (1969) [Soviet Math. Dokl. 10 (5), 1200–1203 (1969)].

    MathSciNet  Google Scholar 

  2. A. Belläiche, “The tangent space in sub-Riemannian geometry”, in Sub-Riemannian Geometry (Birkhäuser, Basel, 1996), vol. 144 of Progress in Math., pp. 1–78.

    Chapter  Google Scholar 

  3. A. Bonfiglioli, E. Lanconelli E., and F. Uguzzoni, Stratified Lie Groups and Potential Theory for their Sub-Laplacians (Springer, Berlin, 2007).

    MATH  Google Scholar 

  4. R. Goodman, “Lifting vector fields to nilpotent Lie groups” J. Math. Pures Appl. (9) 57(1), 77–86 (1978).

    MathSciNet  MATH  Google Scholar 

  5. A. V. Greshnov, “Metrics and tangent cones of uniformly regular Carnot — Carathéodory spaces”, Sibirsk. Mat. Zh. 47(2), 259–292 (2006) [Siberian Math. J. 47 (2), 209–238 (2006)].

    MathSciNet  MATH  Google Scholar 

  6. A. V. Greshnov, “Local approximation of uniformly regular Carnot — Carathéodory quasispaces by their tangent cones”, Sibirsk. Mat. Zh. 48(2), 290–312 (2007) [Siberian Math. J. 48(2), 229–248 (2007)].

    MathSciNet  MATH  Google Scholar 

  7. A. V. Greshnov, “Differentiability of horizontal curves in Carnot—Carathéodory quasispaces”, Sibirsk. Mat. Zh. 49(1), 67–86 (2008) [Siberian Math. J. 49 (1), 53–68 (2008)].

    MathSciNet  MATH  Google Scholar 

  8. A. V. Greshnov, “Applications of the group analysis of differential equations to some systems of noncommuting C1-smooth vector fields”, Sibirsk. Mat. Zh. 50(1), 47–62 (2009) [Siberian Math. J. 50 (1), 37–48 (2009)].

    MathSciNet  MATH  Google Scholar 

  9. A. V. Greshnov, “On applications of the Taylor formula in some quasispaces”, Mat. Tr. 12(1), 3–25 (2009) [Siberian Adv. Math. 20 (3), 164–179 (2010)].

    MathSciNet  MATH  Google Scholar 

  10. M. Gromov, “Carnot — Carathéodory spaces seen from within”, in Sub-Riemannian Geometry (Birkhäuser, Basel, 1996), vol. 144 of Progress in Math., pp. 79–323.

    Chapter  Google Scholar 

  11. A. Koranyi A. and H. M. Reimann, “Foundations for the theory of quasiconformal mappings on the Heisenberg group”, Adv. Math. 111(1), 1–87 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  12. G.A. Margulis and G. D. Mostow, “The differential of a quasi-conformal mapping of a Carnot-Carathéodory space”, Geom. Funct. Anal. 5(2), 402–433 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  13. G. Métivier, “Fonction spectrale et valeurs propres d’une classe d’opérateurs non elliptiques”, Comm. Partial Differ. Equations 1(5), 467–519 (1976).

    Article  MATH  Google Scholar 

  14. J. Mitchell, “On Carnot-Carathéodory metrics”, J. Differential Geom. 21(1), 35–45 (1985).

    MathSciNet  MATH  Google Scholar 

  15. A. Nagel, E. M. Stein, and S. Wainger. “Balls and metrics defined by vector fields I: Basic properties”, Acta Math. 155(1–2), 103–147 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  16. L. V. Ovsyannnikov, Group Analysis of Differential Equations (Nauka, Moscow, 1978) [Academic Press, New York, 1982]

    Google Scholar 

  17. L. S. Pontryagin, Ordinary Differential Equations (Fizmatgiz, Moscow, 1961) [Addison-Wesley, London, 1962].

    MATH  Google Scholar 

  18. L. P. Rothchild and E. M. Stein, “Hypoelliptic differential operators and nilpotent groups”, Acta Math. 137(1), 247–320 (1976).

    Article  MathSciNet  Google Scholar 

  19. E. M. Stein, Harmonic Analysis: Real-Variables Methods, Orthogonality, and Oscillatory Integrals, in Princeton Mathematical Series 43 (Princeton Univ. Press, Princeton, NJ, 1993.)

    Google Scholar 

  20. S. K. Vodop’yanov and A. V Greshnov, “On the differentiability of mappings of Carnot—Carathéodory spaces”, Dokl. Akad. Nauk 389(5), 592–596 (2003) [Doklady Math. 67 (2), 246–250 (2003)].

    MathSciNet  Google Scholar 

  21. S. K. Vodop’yanov and M. B. Karmanova, “Sub-Riemannian geometry for vector fields of minimal smoothness”, Dokl. Akad. Nauk 422(5), 583–588 (2008) [Doklady Math. 78 (2), 737–742 (2008).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia

    A. V. Greshnov

  2. Novosibirsk State University, Novosibirsk, 630090, Russia

    A. V. Greshnov

Authors
  1. A. V. Greshnov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Greshnov.

Additional information

Original Russian Text © A. V. Greshnov, 2010, published in Matematicheskie Trudy, 2010, Vol. 14, No. 1, pp. 70–98.

About this article

Cite this article

Greshnov, A.V. On the generalized triangle inequality for quasimetrics induced by noncommuting vector fields. Sib. Adv. Math. 22, 95–114 (2012). https://doi.org/10.3103/S1055134412020034

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1055134412020034

Keywords

Search

Navigation