In this paper we study the Carnot–Caratheodory metrics on $SU(2)\simeq S^3$, $SO(3)$, and $SL(2)$ induced by their Cartan decomposition and by the Killing form. Besides computing explicitly geodesics and conjugate loci, we compute the cut loci (globally), and we give the expression of the Carnot–Caratheodory distance as the inverse of an elementary function. We then prove that the metric given on $SU(2)$ projects on the so-called lens spaces $L(p,q)$. Also for lens spaces, we compute the cut loci (globally). For $SU(2)$ the cut locus is a maximal circle without one point. In all other cases the cut locus is a stratified set. To our knowledge, this is the first explicit computation of the whole cut locus in sub-Riemannian geometry, except for the trivial case of the Heisenberg group.

MSC codes

  1. 22E30
  2. 49J15
  3. 53C17


  1. left-invariant sub-Riemannian geometry
  2. Carnot–Caratheodory distance
  3. global structure of the cut locus
  4. lens spaces-2pt

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A. Agrachev, Methods of control theory in nonholonomic geometry, in Proceedings of the ICM-94, Birkhäuser, Zürich, 1996, pp. 1473–1483.
A. Agrachev, Compactness for sub-Riemannian length-minimizers and subanalyticity, Rend. Sem. Mat. Univ. Politec. Torino, 56 ( 2001 ), pp. 1 – 12.
A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, Encyclopaedia Math. Sci. 87, Springer-Verlag, Berlin, 2004.
A. Agrachev, Exponential mappings for contact sub-Riemannian structures, J. Dynam. Control Systems, 2 ( 1996 ), pp. 321 – 358.
R. Beals, B. Gaveau, and P. C. Greiner, Hamilton–Jacobi theory and the heat kernel on Heisenberg groups, J. Math. Pures Appl., 79 ( 2000 ), pp. 633 – 689.
A. Bellaïche, The tangent space in sub-Riemannian geometry, in Sub-Riemannian Geometry, A. Bellache and J.-J. Risler, eds., Progr. Math. 144, Birkhäuser, Basel, 1996, pp. 1–78.
A. Boggess, CR Manifolds and the Tangential Cauchy–Riemann Complex, Stud. Adv. Math., CRC Press, Boca Raton, FL, 1991.
B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory, Math. Appl. 40, Springer-Verlag, Berlin, 2003.
U. Boscain, T. Chambrion, and G. Charlot, Nonisotropic 3-level quantum systems: Complete solutions for minimum time and minimum energy, Discrete Contin. Dyn. Syst. Ser. B, 5 ( 2005 ), pp. 957 – 990.
U. Boscain, T. Chambrion, and J.-P. Gauthier, On the $K+P$ problem for a three-level quantum system: Optimality implies resonance, J. Dynam. Control Systems, 8 ( 2002 ), pp. 547 – 572.
U. Boscain and B. Piccoli, Optimal Synthesis for Control Systems on 2-D Manifolds, Math. Appl. 43, Springer-Verlag, Berlin, 2004.
U. Boscain and S. Polidoro, Gaussian estimates for hypoelliptic operators via optimal control, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Mem. Mat. Appl., 18 ( 2007 ), pp. 333 – 342.
A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, AIMS Ser. Appl. Math., AIMS, Springfield, MO, 2007.
R. W. Brockett, Explicitly solvable control problems with nonholonomic constraints, in Proceedings of the 38th Annual IEEE Conference on Decision and Control, Vol. 1, IEEE, Piscataway, NJ, 1999, pp. 13–16.
Y. Chitour, F. Jean, and E. Trélat, Genericity results for singular curves, J. Differential Geom., 73 ( 2006 ), pp. 45 – 73.
Y. Eliashberg, Contact 3-manifolds twenty years since J. Martinet's work, Ann. Inst. Fourier (Grenoble), 42 ( 1992 ), pp. 165 – 192.
B. Gaveau, Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents, Acta Math., 139 ( 1977 ), pp. 95 – 153.
M. Gromov, Carnot–Caratheodory spaces seen from within, in Sub-Riemannian Geometry, A. Bellache and J.-J. Risler, eds., Progr. Math. 144, Birkhäuser, Basel, 1996, pp. 79–323.
L. Hörmander, The analysis of linear partial differential operators, Grundlehren Math. Wiss., Springer-Verlag, Berlin, 1983.
S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure Appl. Math., Academic Press, New York-London, 1978.
V. Jurdjevic, Geometric Control Theory, Cambridge Stud. Adv. Math. 52, Cambridge University Press, Cambridge, UK, 1997.
V. Jurdjevic, Optimal Control, Geometry, and Mechanics, in Mathematical Control Theory, J. Baillieul, and J. C. Willems, eds., Springer, New York, 1999, pp. 227–267.
V. Jurdjevic, Hamiltonian point of view on non-Euclidean geometry and elliptic functions, Systems Control Lett., 43 ( 2001 ), pp. 25 – 41.
L. S. Pontryagin, V. Boltianski, R. Gamkrelidze, and E. Mitchtchenko, The Mathematical Theory of Optimal Processes, Wiley Interscience, New York, London, 1962.
D. Rolfsen, Knots and Links, Publish or Perish, Houston, 1990.
J. R. Weeks, The Shape of Space, Monogr. Textbooks Pure Appl. Math., Marcel Dekker, New York, 1985.

Information & Authors


Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 1851 - 1878
ISSN (online): 1095-7138


Submitted: 25 September 2007
Accepted: 17 February 2008
Published online: 25 June 2008

MSC codes

  1. 22E30
  2. 49J15
  3. 53C17


  1. left-invariant sub-Riemannian geometry
  2. Carnot–Caratheodory distance
  3. global structure of the cut locus
  4. lens spaces-2pt



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