Symmetries of flat rank two distributions and sub-Riemannian structures
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- by Yuri L. Sachkov PDF
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Abstract:
Flat sub-Riemannian structures are local approximations — nilpotentizations — of sub-Riemannian structures at regular points. Lie algebras of symmetries of flat maximal growth distributions and sub-Riemannian structures of rank two are computed in dimensions 3, 4, and 5.References
- A. A. Agrachev and Yu. L. Sachkov, An Intrinsic Approach to the Control of Rolling Bodies, Proceedings of the 38-th IEEE Conference on Decision and Control, vol. 1, Phoenix, Arizona, USA, December 7–10, 1999, 431–435.
- A. A. Agrachëv and A. V. Sarychev, Filtrations of a Lie algebra of vector fields and the nilpotent approximation of controllable systems, Dokl. Akad. Nauk SSSR 295 (1987), no. 4, 777–781 (Russian); English transl., Soviet Math. Dokl. 36 (1988), no. 1, 104–108. MR 906538
- A. Bellaiche, The tangent space in sub-Riemannian Geometry, In Sub-Riemannian Geometry, A. Bellaiche and J.-J. Risler, eds., Birkhäuser, Basel, Swizerland, 1996.
- A. V. Bocharov, V. N. Chetverikov, S. V. Duzhin, N. G. Khor′kova, I. S. Krasil′shchik, A. V. Samokhin, Yu. N. Torkhov, A. M. Verbovetsky, and A. M. Vinogradov, Symmetries and conservation laws for differential equations of mathematical physics, Translations of Mathematical Monographs, vol. 182, American Mathematical Society, Providence, RI, 1999. Edited and with a preface by Krasil′shchik and Vinogradov; Translated from the 1997 Russian original by Verbovetsky [A. M. Verbovetskiĭ] and Krasil′shchik. MR 1670044, DOI 10.1090/mmono/182
- R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldshmidt, and P. A. Griffits, Exterior differential systems, Springer-Verlag, 1984.
- E. Cartan, Les systèmes de Pfaff a cinque variables et les équations aux derivées partielles du second ordre, Ann. Sci. École Normale 27 (1910), 3: 109–192.
- V. Ya. Gershkovich, Engel structures on four dimensional manifolds, Preprint series No. 10, The University of Melbourne, Dept. of Mathematics, 1992.
- Velimir Jurdjevic, Geometric control theory, Cambridge Studies in Advanced Mathematics, vol. 52, Cambridge University Press, Cambridge, 1997. MR 1425878
- J. P. Laumond, Nonholonomic motion planning for mobile robots, LAAS Report 98211, May 1998, LAAS-CNRS, Toulouse, France.
- Alessia Marigo and Antonio Bicchi, Rolling bodies with regular surface: the holonomic case, Differential geometry and control (Boulder, CO, 1997) Proc. Sympos. Pure Math., vol. 64, Amer. Math. Soc., Providence, RI, 1999, pp. 241–256. MR 1654560, DOI 10.1090/pspum/064/1654560
- M. M. Postnikov, Gruppy i algebry Li, “Nauka”, Moscow, 1982 (Russian). Lektsii po geometrii, Semestr V. [Lectures in geometry, Semester V]. MR 685757
- M. Vendittelli, J. P. Laumond, and G. Oriolo, Steering nonholonomic systems via nilpotent approximations: The general two-trailer system, IEEE International Conference on Robotics and Automation, May 10–15, Detroit, MI, 1999.
- A. M. Vershik and V. Ya. 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., Vol. 16; Dynamical Systems VII, Springer-Verlag, 1991.
- D. P. Zhelobenko and A. I. Shtern, Predstavleniya grupp Li, Spravochnaya Matematicheskaya Biblioteka. [Mathematical Reference Library], “Nauka”, Moscow, 1983 (Russian). MR 709598
Additional Information
- Yuri L. Sachkov
- Affiliation: Program Systems Institute, Russian Academy of Sciences, 152140 Pereslavl-Zalessky, Russia
- Email: sachkov@sys.botik.ru
- Received by editor(s): May 4, 2001
- Published electronically: September 22, 2003
- Additional Notes: This work was partially supported by the Russian Foundation for Basic Research, project No. 02-01-00506.
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 356 (2004), 457-494
- MSC (2000): Primary 53C17
- DOI: https://doi.org/10.1090/S0002-9947-03-03342-7
- MathSciNet review: 2022707