Skip to main content
SpringerLink
Log in

Integrability conditions on Engel-type manifolds

  • Published:
Analysis and Mathematical Physics Aims and scope Submit manuscript

Abstract

We introduce the concept of Engel manifold, as a manifold that resembles locally the Engel group, and find the integrability conditions of the associated sub-elliptic system \(Z_1 f = a_1\), \( Z_2 f = a_2\). These are given by \( Z_1^2 a_2 = (Z_1 Z_2 +[Z_1, Z_2]) a_1\), \( Z_2^3 a_1 = (Z_2^2 Z_1 - Z_2 [Z_1, Z_2] - [Z_2, [Z_1, Z_2] ]) a_2\). Then an explicit construction of the solution involving an integral representation is provided, which corresponds to a Poincaré-type lemma for the Engel’s distribution.

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.

Similar content being viewed by others

References

  1. Adams, M.R., Tie, J.: On sub-Riemannian geodesics on the Engel groups: Hamilton’s equation. Mathematische Nachrichten 286(14–15), 1381–1406 (2013)

    MathSciNet  MATH  Google Scholar 

  2. Calin, O., Chang, D.C.: Sub-Riemannian geometry. General theory and examples, encyclopedia of mathematics and its applications, p. 126. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  3. Calin, O., Chang, D.C.: Geometric mechanics on Riemannian manifolds: applications to partial differential equations. Applied and numerical analysis. Birhäuser, Boston (2004)

    Google Scholar 

  4. Calin, O., Chang, D.C., Eastwood, M.: Integrability conditions for the Grushin and Martinet distributions. Bull. IMAS New Ser. 2, 159–168 (2013)

    MathSciNet  Google Scholar 

  5. Calin, O., Chang, D.C., Eastwood, M.: Integrability conditions for Heisenberg and Grushin-type distributions. Anal. Math. Phys. 4, 99–114 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Calin, O., Chang, D.C., Hu, J.: A Poincaré lemma for the Heisenberg group. Adv. Appl. Anal. 60, 90–102 (2014)

    MathSciNet  MATH  Google Scholar 

  7. Hörmander, L.: Hypo-elliptic second order differential equations. Acta. Math. 119, 147–171 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  8. Vershik, A.M., Granichina, G.A.: Reduction of nonholonomic variation problems to isoperimetric ones and connections in principal bundles. Math. Notes Acad. Sci. USSR 49(5), 467–472 (1991)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Eastern Michigan University, Ypsilanti, MI, 48197, USA

    Ovidiu Calin

  2. Department of Mathematics and Statistics, Georgetown University, Washington, DC, 20057, USA

    Der-Chen Chang

  3. Department of Mathematics, Fu Jen Catholic University, New Taipei City, 24205, Taiwan, China

    Der-Chen Chang

  4. Department of Mathematics, Hong Kong University of Sciences and Technology, Kowloon, Hong Kong

    Jishan Hu

Authors
  1. Ovidiu Calin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Der-Chen Chang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jishan Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jishan Hu.

Additional information

This research project is partially supported by Hong Kong RGC competitive earmarked research grants \(\#\)601813, \(\#\)601410.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Calin, O., Chang, DC. & Hu, J. Integrability conditions on Engel-type manifolds. Anal.Math.Phys. 5, 217–231 (2015). https://doi.org/10.1007/s13324-015-0107-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13324-015-0107-3

Keywords

Mathematics Subject Classification

Search

Navigation