Skip to main content
SpringerLink
Log in

The contact mappings of a flat (2,3,5)-distribution

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

Let Ω and Ω′ be open subsets of a flat (2,3,5)-distribution. We show that a C1-smooth contact mapping f: Ω → Ω′ is a \(C^\infty\)-smooth contact mapping. Ultimately, this is a consequence of the rigidity of the associated stratified Lie group. (The Tanaka prolongation of the Lie algebra is of finite type.) The conclusion is reached through a careful study of some differential identities satisfied by components of the Pansu derivative of a C1-smooth contact mapping.

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

Notes

  1. After this paper was submitted, [15] was posted to the arXiv which deals with the C1 case for all rigid groups. The author, Jona Lelmi, uses a striking new characterization of weakly contact vector fields.

References

  1. Austin, A.D., Tyson, J.T.: A new proof of the C∞ regularity of C2 conformal mappings on the Heisenberg group. Colloq. Math. 150(2), 217–228 (2017)

    Article  MathSciNet  Google Scholar 

  2. Baez, J.C., Huerta, J.: G2 and the rolling ball. Trans. Am. Math. Soc. 366(10), 5257–5293 (2014)

    Article  Google Scholar 

  3. Bor, G., Montgomery, R.: G2 and the rolling distribution. Enseign. Math. 55(1–2), 157–196 (2009)

    Article  MathSciNet  Google Scholar 

  4. Capogna, L.: Regularity for quasilinear equations and 1-quasiconformal maps in Carnot groups. Math. Ann. 313(2), 263–295 (1999)

    Article  MathSciNet  Google Scholar 

  5. Capogna, L., Cowling, M.: Conformality and Q-harmonicity in Carnot groups. Duke Math. J. 135(3), 455–479 (2006)

    Article  MathSciNet  Google Scholar 

  6. Cartan, E.: Les systèmes de Pfaff, à cinq variables et les équations aux dérivées partielles du second ordre. Ann. Sci. École Norm. Sup. 3(27), 109–192 (1910)

    Article  Google Scholar 

  7. Christ, M.: Nonexistence of invariant analytic hypoelliptic differential operators on nilpotent groups of step greater than two. In: Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991), volume 42 of Princeton Mathematical Series, pp. 127–145. Princeton University Press, Princeton, NJ, (1995)

  8. Cowling, M., De Mari, F., Korányi, A., Reimann, H.M.: Contact and conformal maps in parabolic geometry. I. Geom. Dedicata 111, 65–86 (2005)

    Article  MathSciNet  Google Scholar 

  9. Cowling, M.G., Ottazzi, A.: Global contact and quasiconformal mappings of Carnot groups. Conform. Geom. Dyn. 19, 221–239 (2015)

    Article  MathSciNet  Google Scholar 

  10. Doubrov, B., Govorov, A.: A new example of a generic 2-distribution on a 5-manifold with large symmetry algebra. arXiv:1305.7297, (2013)

  11. Folland, G.B.: Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat. 13(2), 161–207 (1975)

    Article  MathSciNet  Google Scholar 

  12. Folland, G. B., Stein, E. M.: Hardy Spaces on Homogeneous Groups, volume 28 of Mathematical Notes. Princeton University Press, Princeton (1982)

  13. Hörmander, L.: The analysis of linear partial differential operators. I, volume 256 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin, (1983). Distribution theory and Fourier analysis

  14. Kleiner, B., Müller, S., Xie, X.: Pansu pullback and rigidity of mappings between carnot groups. arXiv:2004.09271, (2020)

  15. Lelmi, J.: On the smoothness of C1-contact maps in C∞-rigid carnot groups. arXiv:2006.06772, (2020)

  16. Liu, Z.: Another proof of the Liouville theorem. Ann. Acad. Sci. Fenn. Math. 38(1), 327–340 (2013)

    Article  MathSciNet  Google Scholar 

  17. Ottazzi, A., Warhurst, B.: Contact and 1-quasiconformal maps on Carnot groups. J. Lie Theory 21(4), 787–811 (2011)

    MathSciNet  MATH  Google Scholar 

  18. Sachkov, Y.L.: Symmetries of flat rank two distributions and sub-Riemannian structures. Trans. Am. Math. Soc. 356(2), 457–494 (2004)

    Article  MathSciNet  Google Scholar 

  19. Sarvas, J.: Ahlfors’ trivial deformations and Liouville’s theorem in Rn. In: Complex analysis Joensuu 1978, volume 747 of Lecture Notes in Mathematical, pp. 343–348. Springer, Berlin, (1979)

  20. Serra Cassano, F.: Some topics of geometric measure theory in Carnot groups. In: Geometry, analysis and dynamics on sub-Riemannian manifolds. Vol. 1, EMS Lecture Notes in Mathematical, pp. 1–121. Zürich, (2016)

  21. Tanaka, N.: On differential systems, graded Lie algebras and pseudogroups. J. Math. Kyoto Univ. 10, 1–82 (1970)

    MathSciNet  MATH  Google Scholar 

  22. Warhurst, B.: Contact and Pansu differentiable maps on Carnot groups. Bull. Aust. Math. Soc. 77(3), 495–507 (2008)

    Article  MathSciNet  Google Scholar 

  23. Willse, T.: Cartan’s incomplete classification and an explicit ambient metric of holonomy G2*. Eur. J. Math. 4(2), 622–638 (2018)

    Article  MathSciNet  Google Scholar 

  24. Yamaguchi, K.: Differential systems associated with simple graded Lie algebras. In: Progress in differential geometry, volume 22 of Advanced Studies in Pure Mathematics, pp. 413–494. Japan, Tokyo, (1993)

Download references

Acknowledgements

The author thanks Alessandro Ottazzi for comments on the Tanaka prolongation of a stratified Lie algebra, Francesco Serra Cassano for elaborating on the proof of Lemma 3.3 and Ben Warhurst for suggesting the Cartan group as suitable test case.

Author information

Authors and Affiliations

  1. Rochester Institute of Technology, 52 Lomb Memorial Drive, Rochester, NY, 14623-5608, USA

    Alex D. Austin

Authors
  1. Alex D. Austin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alex D. Austin.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Austin, A.D. The contact mappings of a flat (2,3,5)-distribution. Ann Glob Anal Geom 60, 143–156 (2021). https://doi.org/10.1007/s10455-021-09767-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-021-09767-4

Keywords

Mathematics Subject Classification

Search

Navigation