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
DOI:
https://doi.org/10.1090/S1079-6762-03-00118-5
Published electronically:
November 13, 2003
MathSciNet review:
2029472
Full-text PDF Free Access
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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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- A. A. Agrachëv, R. V. Gamkrelidze, and A. V. Sarychev, Local invariants of smooth control systems, Acta Appl. Math. 14 (1989), no. 3, 191–237. MR 995286, DOI https://doi.org/10.1007/BF01307214
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as 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.
ags A. A. Agrachev, R. V. Gamkrelidze, A. V. Sarychev, Local invariants of smooth control systems. Acta Appl. Math. 14 (1989), 191–237.
be A. Bellaiche, The tangent space in sub-Riemannian geometry. In: Sub-Riemannian geometry, Birkhäuser, Progress in Math. 144 (1996), 1–78.
bs R. M. Bianchini, G. Stefani, Graded approximations and controllability along a trajectory. SIAM J. Control Optim. 28 (1990), 903–924.
ch W-L. Chow, Über Systeme von linearen partiellen Differentialgleichungen ester Ordnung. Math. Ann. 117 (1940/41), 98–105.
rs L. M. Rothschild, E. M. Stein, Hypoelliptic differential operators and nilpotent groups. Acta Math. 137 (1976), 247–320.
ra 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)
vg A. M. Vershik, V. Ya. Gershkovich, Nonholonomic dynamic systems. Geometry of distributions and variational problems. Springer Verlag, EMS 16 (1987), 5–85.
vg2 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.
vg3 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)
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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
MR Author ID:
190426
Email:
agrachev@ma.sissa.it
A. Marigo
Affiliation:
IAC-CNR, Viale Policlinico 136, Roma, Italy
Email:
marigo@iac.rm.cnr.it
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
