Skip to main content
SpringerLink
Log in

Horizontal Gradient of Polynomial Functions for the Standard Engel Structure on ℝ4

  • Published:
Journal of Dynamical and Control Systems Aims and scope Submit manuscript

Abstract

We investigate the set V f of horizontal critical points of a polynomial function f for the standard Engel structure defined by the 1-forms ω 3 = d x 3 − x 1 d x 2 and ω 4 = d x 4 − x 3 d x 2, endowed with the sub-Riemannian metric \(g_{\text {SR}}=d{x_{1}^{2}}+d{x^{2}_{2}}\). For a generic polynomial, we show that the set Γ f of points in V f , where V f is not transverse to the Engel distribution, does not have a connected component which is contained in a fiber of f. Then, we prove that each trajectory of the horizontal gradient of f approaching the set V f has a limit.

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. Ardentov AA, Sachkov YL. Conjugate points in nilpotent sub-Riemannian problem on the Engel group. J Math Sci 2013;195(3):369–390.

    Article  MathSciNet  MATH  Google Scholar 

  2. Balogh ZM, Holopainen I, Tyson JT. Singular solutions, homogeneous norms, and quasiconformal mappings in Carnot groups. Math. Ann. 2002;324:159–186.

    Article  MathSciNet  MATH  Google Scholar 

  3. Bellaïche A. The tangent space in sub-Riemannian geometry. Progress in Mathematics Vol.144: Birkhäuser Verlag; 1996.

  4. Benedetti R, Risler J-J. Real algebraic and semi-algebraic sets: Hermann; 1991.

  5. Bochnak J, Coste M, Roy M-F. Géométrie algébrique réelle: Springer; 1987.

  6. Chow WL. Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung. Math Ann 1939;117:98–105.

    MathSciNet  Google Scholar 

  7. Coste M. An Introduction to semialgebraic geometry, Dip. Mat. Univ. Pisa, Dottorato di Ricerca in Matematica, Istituti Editoriali e Poligrafici Internazionali, Pisa. 2000. http://perso.univ-rennes1.fr/michel.coste/articles.html.

  8. Dinh ST, Kurdyka K, Orro P. Gradient horizontal de fonctions polynomiales. Annales de l’Institut Fourier 2009;59(5):1999–2042.

    Article  MathSciNet  MATH  Google Scholar 

  9. Engel F. Zur Invariantentheorie der Systeme Pfaff’scher Gleichungen. Leipz Ber Band 1889;41:157–176.

    Google Scholar 

  10. Gershkovich V. Exotic Engel structures on ℝ4. Russian J Math Phys 1995;3(2):207–226.

    MathSciNet  MATH  Google Scholar 

  11. Goresky M, MacPherson R. Stratified Morse theory: Springer; 1988.

  12. Gromov M. 1994. Carnot-Caratheodory spaces seen from within, IHES.

  13. Guillemin V, Pollack A. Differential topology: Prentice-Hall; 1974.

  14. Hardt RM. Semi-algebraic local-triviality in semi-algebraic mappings. Amer J Math 1980;102(2):291–302.

    Article  MathSciNet  MATH  Google Scholar 

  15. Hirsch M. Differential topology: Springer; 1976.

  16. Jean F. Entropy and complexity of a path in sub-Riemannian geometry, ENSTA; 1999.

  17. Kurdyka K, Orro P, Simon S. Semialgebraic sard theorem for generalized critical values. J Diff Geom 2000;56(1):67–92.

  18. S. Łojasiewicz. Ensembles semi-analytiques, I.H.E.S, Bures-sur-Yvette; 1965.

  19. Łojasiewicz S. Sur la géométrie semi- et sous- analytique. Annales de l’institut Fourier 1993;43(5):1575–1595.

    Article  MathSciNet  MATH  Google Scholar 

  20. Łojasiewicz S. Sur le problème de la division. Studia Math 1959;18:87–136.

    MathSciNet  MATH  Google Scholar 

  21. Łojasiewicz S. Sur les trajectoires du gradient d’une fonction analytique, Seminari di Geometria 1982–1983 (Bologna), p. 115–117.

  22. Magnani V. A Blow-up theorem for regular hypersurfaces on nilpotent groups. Manuscripta Math 2003;110(1):55–76.

    Article  MathSciNet  MATH  Google Scholar 

  23. Milnor J. On the Betti numbers of real varieties. Proc Amer Math Soc 1964;15:275–280.

    Article  MathSciNet  MATH  Google Scholar 

  24. Rashevsky PK. Any two points of a totally nonholonomic space may be connected by an admissible line. Uch Zap Ped Inst im Liebknechta Ser Phys Math 1938;2:83–94.

    Google Scholar 

  25. Strichartz R S. Sub-Riemannian geometry. J Diff Geom 1986;24:221–263.

    MathSciNet  MATH  Google Scholar 

  26. Sussmann HJ. Orbits of families of vector fields and integrability of distributions. Trans Amer Math Soc 1973;180:171–188.

    Article  MathSciNet  MATH  Google Scholar 

  27. Thom R. Sur l’homologie des variétés algébriques réelles. Differential and combinatorial topology (A Symposium in Honor of Marston Morse). Princeton: Princeton University Press; 1965, pp. 255–265.

  28. Vogel T. Existence of Engel structures. Ann Math (2) 2009;169(1):79–137.

    Article  MATH  Google Scholar 

  29. Zhitomirskii MY. Normal forms of germs of two-dimensional distributions on ℝ4. (Russian) Funktsional Anal i Prilozhen 1990;24(2):81–82. translation in Funct Anal Appl 1990;24(2):150–152.

Download references

Acknowledgments

This work was supported by the Fields Institute, ANR grant STAAVF (France), LIA Formath Vietnam Project, VAST and Vietnam National Foundation for Science and Technology Development (NAFOSTED) grant 101.04-2014.23.

Author information

Authors and Affiliations

  1. Institute of Mathematics, VAST 18 Hoang Quoc Viet Road, Cau Giay District, 10307, Hanoi, Vietnam

    Si Tiep Dinh

  2. Laboratoire de Mathématiques (LAMA) UMR-5127 CNRS, Campus Scientifique, Université de Savoie, 73376, Le Bourget-du-Lac cedex, France

    Krzysztof Kurdyka

Authors
  1. Si Tiep Dinh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Krzysztof Kurdyka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Krzysztof Kurdyka.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dinh, S.T., Kurdyka, K. Horizontal Gradient of Polynomial Functions for the Standard Engel Structure on ℝ4 . J Dyn Control Syst 22, 15–34 (2016). https://doi.org/10.1007/s10883-014-9242-7

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10883-014-9242-7

Mathematics Subject Classifications (2010)

Search

Navigation