Advertisement

Journal of Dynamical and Control Systems

, Volume 22, Issue 3, pp 413–440 | Cite as

Horizontal Holonomy for Affine Manifolds

  • Boutheina Hafassa
  • Amina Mortada
  • Yacine ChitourEmail author
  • Petri Kokkonen
Article
  • 153 Downloads

Abstract

In this paper, we consider a smooth connected finite-dimensional manifold M, an affine connection ∇ with holonomy group H and Δ a smooth completely non integrable distribution. We define the Δ-horizontal holonomy group \({H^{\;\nabla }_{\Delta }}\) as the subgroup of H obtained by ∇-parallel transporting frames only along loops tangent to Δ. We first set elementary properties of \({H^{\;\nabla }_{\Delta }}\) and show how to study it using the rolling formalism Chitour and Kokkonen (2011). In particular, it is shown that \({H^{\;\nabla }_{\Delta }}\) is a Lie group. Moreover, we study an explicit example where M is a free step-two homogeneous Carnot group with m ≥ 2 generators, and ∇ is the Levi-Civita connection associated to a Riemannian metric on M, and show in this particular case that \({H^{\;\nabla }_{\Delta }}\) is compact and strictly included in H as soon as m ≥ 3.

Keywords

Holonomy group Affine holonomy group Rolling (Development of) manifolds (Sub)Riemannian Geometry Theory of Lie groups. 

Mathematics Subject Classification (2010)

53B05 53C05 53B21 53C17 

Notes

Acknowledgments

The authors thank F. Jean and M. Sigalotti for fruitful discussions and insights regarding the proof of Proposition 5.7.

References

  1. 1.
    Agrachev A, Sachkov Y. An intrinsic approach to the control of rolling bodies. In: Proceedings of the conference on decision and control, Vol. I. Phoenix; 1999. p. 431–435.Google Scholar
  2. 2.
    Agrachev A, Sachkov Y. The orbit theorem and its applications. Encyclopedia ofMathematical Science 2004;87:63–80.MathSciNetGoogle Scholar
  3. 3.
    Bellaiche A. The tangent space in sub-Riemannian geometry progress in mathematics, Vol. 144. Cambridge: Birkhauser; 1996.Google Scholar
  4. 4.
    Besse A. Einstein manifolds. Berlin Heidelberg New York: Springer; 2007.zbMATHGoogle Scholar
  5. 5.
    Bonfiglioli A, Lanconelli E, Uguzzoni F. Stratified lie groups and potential theory for their Sub-Laplacians. Springer Monographs in Mathematics. 2007.Google Scholar
  6. 6.
    Cartan E. La géométrie des espaces de Riemann. Mémorial des Sciences Mathématiques 1925;9:1–61.Google Scholar
  7. 7.
    Chitour Y. A continuation method for motion-planning problems. ESAIM-COCV. 2006;12:139–168.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chitour Y, Godoy Molina M, Kokkonen P. The rolling problem: overview and challenges. Geometric control theory and sub-Riemannian geometry. Cham: Springer; 2014, pp. 103–122. Springer INdAM Ser., 5.Google Scholar
  9. 9.
    Chitour Y, Kokkonen P. Rolling manifolds: intrinsic formulation and controllability. 2011. Preprint, arXiv:1011.2925v2.
  10. 10.
    Chitour Y, Kokkonen P. Rolling manifolds and controllability: the 3D case. 2015. Accepted for publication in Mémoires de la SMF.Google Scholar
  11. 11.
    Chitour Y, Kokkonen P. Rolling manifolds on space forms. Ann I H Poincaré - AN. 2012;29:927–954.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Drager DL, Lee JM, Park E, Richardson K. Smooth distributions are finitely generated. Ann Glob Anal Geom. 2012.Google Scholar
  13. 13.
    Falbel E, Gorodsky C, Rumin M. Holonomy of sub-Riemannian manifolds. Int J Math. 1997;8:317–344.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Godoy Molina M, Grong E, Markina I, Leite FS. An intrinsic formulation of the rolling manifolds problem. J Dyn Control Syst. 2012;18.Google Scholar
  15. 15.
    Goto M. On an arcwise connected subgroup of a lie group. Proc Am Math Soc 1969;20(1).Google Scholar
  16. 16.
    Harms P. The poincaré lemma in Subriemannian geometry. 2012. arXiv:1211.3531.
  17. 17.
    Hilgert J, Neeb KH. Structure and geometry of lie groups. Berlin Heidelberg New York: Springer; 2011.zbMATHGoogle Scholar
  18. 18.
    Jean F. Control of nonholonomic systems: from Sub-Riemannian geometry to motion planning. Springer Briefs in Mathematics. 2014.Google Scholar
  19. 19.
    Jurdjevic V. Geometric control theory. Cambridge: Cambridge University Press; 1997.zbMATHGoogle Scholar
  20. 20.
    Joyce D. Riemannian holonomy groups and calibrated geometry. New York: Oxford University Press Inc.; 2007.zbMATHGoogle Scholar
  21. 21.
    Kokkonen P. A characterization of isometries between Riemannian manifolds by using development along geodesic triangles. Archivum Mathematicum, Tomus 48, Number 3. 2012.Google Scholar
  22. 22.
    Kokkonen P. Rolling of manifolds without spinning. J Dyn Control Syst. 2013;19(1).Google Scholar
  23. 23.
    Kobayashi S, Nomizu K. Foundations of differential geometry, Vol. I. New York: Wiley; 1996.Google Scholar
  24. 24.
    Mortada A, Kokkonen P, Chitour Y. Rolling manifolds of different dimensions. 2014. Accepted for publication in Acta Applicandae Mathematicae.Google Scholar
  25. 25.
    Ozeki H. Infinitesimal holonomy groups of bundle connections. J Nagoya Math. 1956;10:105–123.MathSciNetzbMATHGoogle Scholar
  26. 26.
    Postnikov MM. Geometry VI - Riemannian geometry, Encyclopedia of Mathematical Sciences. Berlin Heidelberg New York: Springer; 2001.zbMATHGoogle Scholar
  27. 27.
    Rifford L. Subriemannian geometry and optimal transport, Springer Briefs in Mathematics; 2014.Google Scholar
  28. 28.
    Sakai T. Riemannian geometry. Translations of mathematical monographs, Vol. 149. Providence: American Mathematical Society; 1996.Google Scholar
  29. 29.
    Sharpe RW. Differential geometry: Cartan’s generalization of Klein’s Erlangen program, vol. 166, Graduate texts in mathematics. New York: Springer; 1997.Google Scholar
  30. 30.
    Sussmann H. Smooth distributions are globally finitely spanned. Analysis and design of nonlinear control systems. 2008.Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Boutheina Hafassa
    • 1
    • 2
  • Amina Mortada
    • 1
    • 3
  • Yacine Chitour
    • 1
    Email author
  • Petri Kokkonen
    • 4
  1. 1.Université Paris-Sud 11, CNRS, SupélecGif-sur-YvetteFrance
  2. 2.Département de MathématiquesUniversité de Tunis El Manar, FST, MISTMTunisTunisia
  3. 3.Université Libanaise, LNCSR ScholarBeirutLebanon
  4. 4.HelsinkiFinland

Personalised recommendations