Exponential mappings for contact sub-Riemannian structures. (English) Zbl 0941.53022

Author’s abstract: “On a sub-Riemannian manifolds any neighborhood of any point contains geodesics which are not length minimizers; the closures of the cut and the conjugate loci of any point \(q\) contain \(q\). We study this phenomenon in the case of a contact distribution, essentially in the lowest possible dimension 3, where we extract differential invariants related to the singularities of the cut and the conjugate loci near \(q\) and give a generic classification of these singularities”.


53C17 Sub-Riemannian geometry
58K05 Critical points of functions and mappings on manifolds
53D10 Contact manifolds (general theory)
53C20 Global Riemannian geometry, including pinching
53C22 Geodesics in global differential geometry
Full Text: DOI


[1] A. A. Agrachev, Methods of control theory in nonholonomic geometry.Proc. ICM-94, Zürich, Birkhäuser, 1995, 1473–1483. · Zbl 0848.93012
[2] A. A. Agrachev, El-H. C. El Alaoui, J. -P. Gauthier, and I. Kupka, Generic sub-Riemannian metrics onR 3 Compt. Rend. Acad. Sci. Ser. 1 322 (1996), 377–384. · Zbl 0843.53025
[3] A. A. Agrachev and R. V. Gamkrelidze, Exponential representation of flows and chronological calculus. (Russian)Mat. sb. 107 (1978) 467–532. (English translation: Math. USSR Sb.29 (1979), 727–785.) · Zbl 0408.34044
[4] A. A. Agrachev, Symplectic methods for optimization and control (to appear in: Geometry of Feedback and Optimal Control, B. Jakubczyk and W. Respondek. Eds,Marcel Dekker).
[5] A. A. Agrachev, R. V. Gamkrelidze, and A. V. Sarychev, Local invariants of smooth control systems.Acta Appl. Math. 14 (1989), 191–237. · Zbl 0681.49018 · doi:10.1007/BF01307214
[6] A. A. Agrachev, S. Stefani, and P. L. Zezza, Strong minima in optimal control (in preparation). · Zbl 0936.49014
[7] V. I. Arnold, Mathematical methods in classical mechanics, Third edition,Nauka, Moscow, 1989.
[8] R. W. Brockett, Control theory and singular Riemannian geometry. In: New Directions in Applied Mathematics, P. J. Hilton and G. S. Young, Eds.Springer Verlag, 1981. · Zbl 0505.93064
[9] Ge Zhong, Horizontal path space and Carnot-Caratheodory metrics.Pac. J. Math. 161, (1993), 255–286. · Zbl 0797.49033
[10] M. Golubitsky and V. Guillemin, Stable mappings and their singularities.Springer Verlag, New York, 1973. · Zbl 0294.58004
[11] M. Gromov, Carnot-Caratheodory spaces seen from within.Preprint IHES/M/94/6, 1994
[12] R. Montgomery, The isoholonomic problem and some applications,Commun. Math. Phys. 128, (1990), 565–592. · Zbl 0703.53024 · doi:10.1007/BF02096874
[13] A. V. Sarychev, The index of the second variation of a control system.Mat. Sb. 113 (1980) 464–486. (English translation:Math. USSR Sb. 41 (1982) 383–401.)
[14] A. M. Vershik and V. Y. Gershkovich, Nonholonomic dynamical systems. Geometry of distributions and variational problems. (Russian) In: Itogi Nauki i Tekhniki: Sovremennye Problemy Matematiki, Fundamentalnye Napravleniya, Vol. 16,VINITI, Moscow, 1987, 5–85. (English translation) in:Encyclopedia of Math. Sci. 16, Dynamical Systems 7,Springer Verlag). · Zbl 0797.58007
[15] H. Whitney, On singularities of mappings of Euclidean spaces.Ann. math. 62 (1955). · Zbl 0068.37101
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