Journal of Dynamical and Control Systems

, Volume 1, Issue 1, pp 49–90 | Cite as

A survey of singular curves in sub-Riemannian geometry

  • R. Montgomery


Sub-Riemannian geometry is the geometry of a distribution ofk-planes on an-dimensional manifold with a smoothly varying inner product on thek-planes. Singular curves are singularities of the space of paths tangent to the distribution and joining two fixed points. This survey is devoted to the singular curves, which can be length minimizing geodesics, independent of the choice of inner product.

1991 Mathematics Subject Classification

49K30 53C22 


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  1. 1.
    A.A. Agrachev and A.V. Sarychev, Abnormal sub-Riemannian geodesics: Morse index and rigidity.Preprint, 1993.Google Scholar
  2. 2.
    W. Ambrose and I.M. Singer, A theorem on holonomy.Trans. Am. Math. Soc. 75 (1953), 428–453.Google Scholar
  3. 3.
    A. Bejancu, Geometry of CR-submanifolds.D. Reidel, Dordrecht, Holland, 1986.Google Scholar
  4. 4.
    J.-M. Bismut, Large deviations and the Malliavin calculus.Birkhauser, 1984.Google Scholar
  5. 5.
    G.A. Bliss, Lectures on calculus of variations.Univ. of Chicago Press, 1946.Google Scholar
  6. 6.
    R.W. Brockett, Nonlinear control theory and differential geometry.Proc. of the Int. Congress of Mathematicians, Warszawa, 1983.Google Scholar
  7. 7.
    R.L. Bryant, S.S. Chern, R.B. Gardner, H.L. Goldschmidt, and P.A. Griffiths, Exterior differential systems, Vol. 18.MSRI Publications, Springer-Verlag, 1991.Google Scholar
  8. 8.
    R. Bryant and L. Hsu, Rigidity of integral curves of rank two distributions.Invent. Math. 114 (1993), 435–461.Google Scholar
  9. 9.
    E. Cartan, Lès systemes de Pfaff a cinque variables et lès equations aux derivees partielles du second ordre.Ann. Sci. Ècole Normale 27 (1910), No. 3, 109–192.Google Scholar
  10. 10.
    L. Chaves A finiteness theorem for fat bundles.Topology. (1994), Aug., 493–499.Google Scholar
  11. 11.
    W.L. Chow, Uber Systeme van Linearen partiellen Differentialgleichungen erster Ordnung.Math. Ann. 117 (1939), 98–105.Google Scholar
  12. 12.
    M. Christ, Analytic hypoellipticity breaks down for weakly pseudoconvex Reinhardt domains.Preprint, UCLA, 1991.Google Scholar
  13. 13.
    K.J. Falconer, The geometry of fractal sets.Cambridge Univ. Press, 1985.Google Scholar
  14. 14.
    Zhong Ge, Horizontal path space and Carnot-Caratheodory metrics.Pac. J. Math. 161 (1993), 255–286.Google Scholar
  15. 15.
    —, Caustics in optimal control: an example when the symmetry is broken.Lect. Appl. Math. 29 (1993), 203–212.Google Scholar
  16. 16.
    Zhong Ge, On the cut points and conjugate points ina constrained variational problem.Fields Inst. Commun. 1 (1993).Google Scholar
  17. 17.
    —, On a constrained variational problem and the space of horizontal paths.Pac. J. Math. 149 (1991), 61–94.Google Scholar
  18. 18.
    V. Gershkovich, Engel structures on four-dimensional manifolds.Preprint, Univ. of Melbourne, 1993.Google Scholar
  19. 19.
    M. Gromov, J. Lafontaine, and P. Pansu, Structures metriques pour les varietes Riemanniennes.Paris, Cedic, 1981.Google Scholar
  20. 20.
    M. Gromov, Carnot-Caratheodory spaces seen from within.IHES preprint M/94/6, 1994, 221 pp.Google Scholar
  21. 21.
    A. Guichardet, On rotation and vibration motions of molecules.Ann. Inst. H. Poincare, Phys. Theor. 40 (1984), No. 3, 329–342.Google Scholar
  22. 22.
    V. Guillemin and A. Uribe, The trace formula for vector bundles.Bull. Am. Math. Soc. 14 (1986), No. 2, 222–224.Google Scholar
  23. 23.
    V. Guillemin, Some spectral properties of periodic potentials. inLect. Notes Math. 1256 (1987),Springer, 192–214.Google Scholar
  24. 24.
    —, The Laplace operator on thenth tensor power of a line bundle: eigenvalues which are uniformly bounded inn.Asympt. Anal. 1 (1988), 105–113.Google Scholar
  25. 25.
    U. Hamenstädt, Some regularity theorems for Carnot-Caratheodory metricsJ. Differ. Geom. 32 (1990), 819–850.Google Scholar
  26. 26.
    —, Zur Theorie von Carnot-Caratheodory Metriken und ihren Anwendungen.Bonner Math. Schriften, Univ. Bonn, (1987), No. 180, 1–64.Google Scholar
  27. 27.
    R. Hermannn, Some differential geometric aspects of the Lagrange variational problem.Indiana Math. J. (1962), 634–673.Google Scholar
  28. 28.
    L. Hsu, Calculus of variations via the Griffiths formalism.J. Differ. Geom. 36 (1991), No. 3, 551–591.Google Scholar
  29. 29.
    Th. Kaluza, Zum unitatsproblem der physik,Berlin Berichte (1921), 966 pp.Google Scholar
  30. 30.
    R. Kerner, Generalization of the Kaluza-Klein theory for an arbitrary non-Abelian group.Ann. Inst. Henri Poincare 9 (1968), 143–152.Google Scholar
  31. 31.
    I. Kupka, Abnormal extremals.Preprint, 1992.Google Scholar
  32. 32.
    W-S. Liu and H.J. Sussmann, Shortest paths for sub-Riemannian metrics on rank two distributions.Preprint, Trans. Am. Math. Soc., 1994 (to appear).Google Scholar
  33. 33.
    J. Martinet, Sur les singularites des formes differentialles.Ana. Insp. (Grenoble) 20 (1970), No. 1, 95–198.Google Scholar
  34. 34.
    J. Mitchell, On Carnot-Caratheodory metrics.J. Differ. Geom. 21 (1985), 34–45.Google Scholar
  35. 35.
    R. Montgomery, Abnormal minimizers.SIAM J. Control Optim. (1994) (to appear).Google Scholar
  36. 36.
    —, The isoholonomic problem and some applications.Commun. Math. Phys.,128 (1990), 565–592.Google Scholar
  37. 37.
    R. Montgomery, Singular extremals on Lie groups,MCSS (1995) (to appear).Google Scholar
  38. 38.
    —, Gauge theory and control theory. In Nonholonomic motion planning. J. Canny and Z. Li, eds.,Kluwer Acad., Norwell, MA, USA, 1993, 343–378.Google Scholar
  39. 39.
    —, Gauge theory and the falling cat. In Dynamics and control of mechanical systems. M. Enos, ed., Fields Inst. Communications Series, Vol.1,AMS Pub., Providence, RI, 1993, 193–218.Google Scholar
  40. 40.
    —, Generic distributions and finite dimensional Lie algebras.J. Differ. Equ. 103 (1993), 387–393.Google Scholar
  41. 41.
    R. Montgomery, Hearing the zero locus of a magnetic field.Preprint, 1994.Google Scholar
  42. 42.
    M. Morse and S. Myers, The problems of Lagrange and Mayer with variable endpoints.Proc. Am. Acad. 66 (1931), No. 6, 236–253.Google Scholar
  43. 43.
    R. Murray, Nilpotent bases for a class of non-integrable distributions with applications to trajectory generation for nonholonomic systems. CDS Technical Memorandum, No. CIT-CDS 92-002,Calif. Inst. of Tech., 1993.Google Scholar
  44. 44.
    A. Nagel, E. M. Stein, and S. Wainger, Balls and metrics defined by vector fields.Acta Math. 155 (1985), 103–147.Google Scholar
  45. 45.
    P. Pansu, Metriques de Carnot-Caratheodory et quasiisometries des espaces symetriques de rang un.Ann. Math. 129 (1989), 1–60.Google Scholar
  46. 46.
    L. S. Pontrjagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The mathematical theory of optimal processes.Wiley Interscience, 1962.Google Scholar
  47. 47.
    C.B. Rayner, The exponential map for the Lagrange problem on differentiable manifolds.Philos. Trans. R. Soc. London, Ser. A, Math. and Phys. Sci. 262 (1967), No. 1127, 299–344.Google Scholar
  48. 48.
    L. P. Rothschild and E.M. Stein, Hypoelliptic differential operators and nilpotent groups.Acta Math. 137 (1976), 247–320.Google Scholar
  49. 49.
    A. Shapere and F. Wilczek, Self-propulsion at low Reynolds number.Phys. Rev. Lett. 58 (1987), 2051–2054.Google Scholar
  50. 50.
    —, Geometric phases in physics.World Scientific, Singapore, 1989.Google Scholar
  51. 51.
    E.D. Sontag, Mathematical control theory.Springer-Verlag, New York, 1990.Google Scholar
  52. 52.
    S. Sternberg, On minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang-Mills field.Proc. Natl. Acad. Sci. USA 74 (1977), 5253–5254.Google Scholar
  53. 53.
    R. Strichartz, Sub-Riemannian geometry.J. Differ. Geom. 24 (1983), 221–263.Google Scholar
  54. 54.
    —, Corrections to “Sub-Riemanniand geometry”.J. Differ. Geom. 30 (1989), No. 2, 595–596.Google Scholar
  55. 55.
    A.M. Vershik and V. Ya Gerhskovich, An estimate of the functional dimension of the space of orbits of germs of generic distributions.Math. USSR, Zametki 44:45 (1988), 596–603.Google Scholar
  56. 56.
    —, Nonholonomic dynamical systems, geometry of distributions and variational problems. In Dynamical Systems VII. Ed. V.I. Arnol'd and S.P. Novikov, Vol. 16 of the Encyclopaedia of Mathematical Sciences series,Springer-Verlag, NY, 1994. (Russian original 1987.)Google Scholar
  57. 57.
    A. Weinstein, Fat bundles and symplectic manifolds.Adv. Math. 37 (1980), 239–250.Google Scholar
  58. 58.
    —, A universal phase space for a particle in a Yang-Mills field.Lett. Math. Phys. 2 (1978), 417–420.Google Scholar
  59. 59.
    S.K. Wong, Field and particle equations for the classical Yang-Mills field and particles with isotopic spin.Nuovo Cimento 65a (1970), 689–693.Google Scholar
  60. 60.
    L.C. Young, Lectures on the calculus of variations and optimal control theory.Chelsea, 1980.Google Scholar
  61. 61.
    M. Ya. Zhitomirskii, Typical singularities of differential 1-forms and Pfaffian equations. Transl. of Math. Monographs series, Vol. 113,A.M.S., Providence, RI, 1992.Google Scholar
  62. 62.
    —, Singularities and normal forms of odd-dimensional Pfaff equations. (Russian)Funkts. Anal. Ego Prilozhen. 23 (1989), No. 1, 70–71.Google Scholar
  63. 63.
    —, Normal forms of germs of 2-dimensional distributions in ℝ4. (Russian)Funkts. Anal. Ego Prilozhen. 24 (1990), No. 2, 81–82, English translation:Funct. Anal. Appl. 24, (1990).Google Scholar
  64. 64.
    J.W. Zwanziger, M. Koenig, and A. Pines, Berry's phase. InAnnual Review of Chemistry, 1990.Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • R. Montgomery
    • 1
  1. 1.Mathematics Dept.UCSCSanta CruzUSA

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