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Nonholonomic tangent spaces: intrinsic construction and rigid dimensions

Authors: A. Agrachev and A. Marigo
Journal: Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 111-120
MSC (2000): Primary 58A30; Secondary 58K50
Published electronically: November 13, 2003
MathSciNet review: 2029472
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Abstract: A nonholonomic space is a smooth manifold equipped with a bracket generating family of vector fields. Its infinitesimal version is a homogeneous space of a nilpotent Lie group endowed with a dilation which measures the anisotropy of the space. We give an intrinsic construction of these infinitesimal objects and classify all rigid (i.e. not deformable) cases.

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  • 1. A. A. Agrachev, A. V. Sarychev, Filtrations of a Lie algebra of vector fields and nilpotent approximation of control systems. Dokl. Akad. Nauk SSSR 295 (1987); English transl., Soviet Math. Dokl. 36 (1988), 104-108. MR 88j:93015
  • 2. A. A. Agrachev, R. V. Gamkrelidze, A. V. Sarychev, Local invariants of smooth control systems. Acta Appl. Math. 14 (1989), 191-237. MR 90i:93033
  • 3. A. Bellaiche, The tangent space in sub-Riemannian geometry. In: Sub-Riemannian geometry, Birkhäuser, Progress in Math. 144 (1996), 1-78. MR 98a:53108
  • 4. R. M. Bianchini, G. Stefani, Graded approximations and controllability along a trajectory. SIAM J. Control Optim. 28 (1990), 903-924. MR 91d:93006
  • 5. W-L. Chow, Über Systeme von linearen partiellen Differentialgleichungen ester Ordnung. Math. Ann. 117 (1940/41), 98-105. MR 1:313d
  • 6. L. M. Rothschild, E. M. Stein, Hypoelliptic differential operators and nilpotent groups. Acta Math. 137 (1976), 247-320. MR 55:9171
  • 7. P. K. Rashevskii, About connecting two points of a completely nonholonomic space by an admissible curve. Uch. Zapiski Ped. Inst. Libknechta, No. 2 (1938), 83-94. (Russian)
  • 8. A. M. Vershik, V. Ya. Gershkovich, Nonholonomic dynamic systems. Geometry of distributions and variational problems. Springer Verlag, EMS 16 (1987), 5-85. MR 89f:58007
  • 9. A. M. Vershik, V. Ya. Gershkovich, A bundle of nilpotent Lie algebras over a nonholonomic manifold (nilpotenization). Zap. Nauch. Sem. LOMI 172 (1989), 3-20. English transl., J. Soviet Math. 59 (1992), 1040-1053. MR 91a:58012
  • 10. A. M. Vershik, V. Ya. Gershkovich, Estimation of the functional dimension of the orbit space of germs of distributions in general position. Mat. Zametki 44 (1988), no. 5, 596-603, 700. English transl., Math. Notes 44 (1988), no. 5-6, 806-810 (1989) MR 90g:58002

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Additional Information

A. Agrachev
Affiliation: Steklov Mathematical Institute, Moscow, Russia
Address at time of publication: SISSA, Via Beirut 2–4, Trieste, Italy

A. Marigo
Affiliation: IAC-CNR, Viale Policlinico 136, Roma, Italy

Keywords: Nonholonomic system, nilpotent approximation, Carnot group
Received by editor(s): March 25, 2003
Published electronically: November 13, 2003
Communicated by: Svetlana Katok
Article copyright: © Copyright 2003 American Mathematical Society

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