Journal of Dynamical and Control Systems

, Volume 18, Issue 1, pp 21–44 | Cite as

Sub-Riemannian structures on 3D lie groups

  • A. AgrachevEmail author
  • D. Barilari

We give a complete classification of left-invariant sub-Riemannian structures on three-dimensional Lie groups in terms of the basic differential invariants. As a consequence, we explicitly find a sub-Riemannian isometry between the nonisomorphic Lie groups SL(2) and A +(\( \mathbb{R} \)) × S 1, where A +(\( \mathbb{R} \)) denotes the group of orientation preserving affine maps on the real line.

Key words and phrases

Sub-Riemannian geometry Lie groups left-invariant structures 

2000 Mathematics Subject Classification

53C17 22E30 49J15 


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  1. 1.
    A. Agrachev, D. Barilari, and U. Boscain. Introduction to Riemannian and sub-Riemannian geometry. files/notes.html.
  2. 2.
    ______, On the Hausdorff volume in sub-Riemannian geometry. Calculus of Variations and Partial Differential Equations  10.1007/s00526-011-0414-y (2011), p. 1–34.
  3. 3.
    A. Agrachev, U. Boscain, J.-P. Gauthier, and F. Rossi. The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups. J. Funct. Anal. 256 (2009), No. 8, 2621–2655.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    A. A. Agrachev. Exponential mappings for contact sub-Riemannian structures. J. Dynam. Control Systems 2 (1996), No. 3, 321–358.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    A. A. Agrachev, G. Charlot, J. P. A. Gauthier, and V. M. Zakalyukin. On sub-Riemannian caustics and wave fronts for contact distributions in the three-space. J. Dynam. Control Systems 6 (2000), No. 3, 365–395.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    A. A. Agrachev and Yu. L. Sachkov. Control theory from the geometric viewpoint. Encycl. Math. Sci. 87, Springer-Verlag, Berlin (2004).Google Scholar
  7. 7.
    A. Bellaïche. The tangent space in sub-Riemannian geometry. J. Math. Sci. 83 (1997), No. 4, 461–476.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    U. Boscain and F. Rossi. Invariant Carnot–Carathéodory metrics on S3, SO(3), SL(2), and lens spaces. SIAM J. Control Optim. 47 (2008), No. 4, 1851–1878.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    É. Cartan. Sur la géométrie pseudo-conforme des hypersurfaces de l’espace de deux variables complexes, II. Ann. Scu. Norm. Sup. Pisa Cl. Sci. (2) 1 (1932), No. 4, 333–354.MathSciNetzbMATHGoogle Scholar
  10. 10.
    É. Cartan. Sur la géométrie pseudo-conforme des hypersurfaces de l’espace de deux variables complexes. Ann. dMat. Pura Appl. 11 (1933), No. 1, 17–90.MathSciNetCrossRefGoogle Scholar
  11. 11.
    W.-L. Chow. Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung. Math. Ann. 117 (1939), 98–105.MathSciNetCrossRefGoogle Scholar
  12. 12.
    E. Falbel and C. Gorodski. Sub-Riemannian homogeneous spaces in dimensions 3 and 4. Geom. Dedicata 62 (1996), No. 3, 227–252.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    V. Gershkovich and A. Vershik. Nonholonomic manifolds and nilpotent analysis. J. Geom. Phys. 5 (1988), No. 3, 407–452.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    M. Gromov. Carnot–Carathéodory spaces seen from within. Progr. Math. 144, Birkhäuser, Boston (1996), 79–323.Google Scholar
  15. 15.
    N. Jacobson. Lie algebras. Interscience Publishers, New York–London (1962).Google Scholar
  16. 16.
    I. Moiseev and Yu. L. Sachkov. Maxwell strata in sub-Riemannian problem on the group of motions of a plane. ESAIM Control Optim. Calc. Var. 16 (2010), 380–399.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    R. Montgomery. A tour of subriemannian geometries, their geodesics and applications. Math. Surv. Monogr. 91, Am. Math. Soc. Providence, Rhode Island (2002).Google Scholar
  18. 18.
    T. Nagano. Linear differential systems with singularities and an application to transitive Lie algebras. J. Math. Soc. Jpn. 18 (1966), 398–404.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    P. Rashevsky. Any two points of a totally nonholonomic space may be connected by an admissible line. Uch. Zap. Liebknecht Ped. Inst. 2 (1938), 83–84.Google Scholar
  20. 20.
    Yu. L. Sachkov. Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane. ESAIM Control Optim. Calc. Var. 16 (2010), 1018–1039.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    R. S. Strichartz. Sub-Riemannian geometry. J. Differ. Geom. 24 (1986), No. 2, 221–263.MathSciNetzbMATHGoogle Scholar
  22. 22.
    _______, Corrections to: “Sub-Riemannian geometry.” J. Differ. Geom. 30 (1989), No. 2, 595–596.MathSciNetGoogle Scholar
  23. 23.
    H. J. Sussmann. An extension of a theorem of Nagano on transitive Lie algebras. Proc. Am. Math. Soc. 45 (1974), 349–356.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    ________, Lie brackets, real analyticity and geometric control. Progr. Math. 27, Birkhäuser, Boston (1983), 1–116.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.SISSATriesteItaly
  2. 2.Steklov Mathematical InstituteMoscowRussia

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