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Agrachev, A.; Bonnard, B.; Chyba, M.; Kupka, I.

Sub-Riemannian sphere in Martinet flat case. (English) Zbl 0902.53033

ESAIM, Control Optim. Calc. Var. 2, 377-448 (1997).
The authors study the sub-Riemannian geometry \((\mathbb{R}^3, D,g)\), where \(D: y^2 dx=2dz\) is the Martinet distribution and \(g\) is an analytic Riemannian metric on \(D \). The metric \(g\) can be reduced to the flat case or to a one-parameter deformation of the flat case. In this context, the authors discover a parametrization of geodesics using elliptic integrals and derive the exponential map, the wave front, the conjugate and cut loci, the sub-Riemannian sphere, etc. An appendix contains numerical and graphical results concerning the conjugate locus and the sub-Riemannian sphere.
Reviewer: C.Udrişte (Bucureşti)

Cited in 5 Reviews
Cited in 41 Documents

MSC:

53C22 Geodesics in global differential geometry
49L99 Hamilton-Jacobi theories
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)

Keywords:

sub-Riemannian geometry; Martinet distribution; cut locus; conjugate loci
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Full Text: DOI EuDML

References:

[1] A. Agrachev, A. V. Sarychev: Strong minimality of abnormal geodesics for 2-distributions, Journal of Dynamical and control Systems, 2, 1995, 139-176. Zbl0951.53029 MR1333769 · Zbl 0951.53029 · doi:10.1007/BF02254637
[2] A. Agrachev: Exponential mappings for contact sub-Riemannian structures, Journal of dynamical and Control Systems, 2, 1996, 321-358. Zbl0941.53022 MR1403262 · Zbl 0941.53022 · doi:10.1007/BF02269423
[3] A. Agrachev: Any smooth simple H1-local length minimizer in the Carnot-Caratheodory space is a C0-local minimizer, Preprint of Laboratoire de Topologie, Dijon, 1996.
[4] V. I. Arnold: Méthodes mathématiques pour la mécanique classique, Éditions MIR, Moscou, 1976. Zbl0385.70001 MR474391 · Zbl 0385.70001
[5] G.A. Bliss: Lectures on the calculus of variations, The University of Chicago Press, 1946. Zbl0063.00459 MR17881 · Zbl 0063.00459
[6] B. Bonnard: Feedback equivalence for nonlinear systems and the time optimal control problem, SIAM J. on Control and Opt., 29, 1991, 1300-1321. Zbl0744.93033 MR1132184 · Zbl 0744.93033 · doi:10.1137/0329067
[7] B. Bonnard, M. Chyba: Exponential mapping, sphere and waves front in SR-geometry: the generic integrable Martinet case, Preprint of Laboratoire de Topologie, Dijon, 1997.
[8] B. Bonnard, M. Chyba, H. Heutte: Contrôle optimal géométrique appliqué, Preprint of Laboratoire de Topologie, Dijon, 1995.
[9] B. Bonnard, M. Chyba, I. Kupka: Non-integrable geodesics in SR Martinet geometry, in Proceedings AMS conference, Boulder, 1997. Zbl0963.53015 · Zbl 0963.53015
[10] B. Bonnard, M. Chyba, E. Trélat: Sub-Riemannian geometry: one parameter deformation of the Martinet flat case, to appear in Journal of Dynamical and Control Systems. Zbl0967.53020 MR1605346 · Zbl 0967.53020
[11] R. W. Brockett: Control theory and singular Riemannian geometry, in New directions in applied Math., Springer-Verlag, New-York, 1981. Zbl0483.49035 MR661282 · Zbl 0483.49035
[12] E. Cartan: Leçons sur la géométrie des espaces de Riemann, Ed. J. Gabay, Paris, 1988. Zbl0060.38101 MR1191392 · Zbl 0060.38101
[13] H. Davis: Introduction to non linear differential and integral equation, Dover, New-York, 1962. Zbl0106.28904 · Zbl 0106.28904
[14] J. Dieudonné: Calcul Infinitésimal, Hermann, Paris, 1980. Zbl0497.26004 MR226971 · Zbl 0497.26004
[15] M. Do Carmo: Riemannian geometry, Birkhauser, Boston, 1992. Zbl0752.53001 MR1138207 · Zbl 0752.53001
[16] L. V .D. Dries, A. Macintyre, D. Marker: The elementary theory of restricted analytic fields with exponentiation, Annals of Mathematics, 140, 1994, 183-205. Zbl0837.12006 MR1289495 · Zbl 0837.12006 · doi:10.2307/2118545
[17] C. El Alaoui, J. P. Gauthier, I. Kupka: Small sub-Riemannian balls on R3, Journal of dynamical and Control Systems, 2, 1996, 359-421. Zbl0941.53024 MR1403263 · Zbl 0941.53024 · doi:10.1007/BF02269424
[18] R. Gérard, H. Tahora: Singular nonlinear PDE, Vieweg-Verlag, Germany, 1996.
[19] J. Gregory: Quadratic form theory and differential equation, Academic Press, New-York, 1980. Zbl0468.15015 MR599362 · Zbl 0468.15015
[20] U. Hamenstadt: Some regularity theorem for Carnot-Caratheodory metries, J. Differential geometry, 32, 1991, 819-850. Zbl0687.53041 MR1078163 · Zbl 0687.53041
[21] F. John: Partial differential equations, Springer-Verlag, New-York, 1971. Zbl0209.40001 · Zbl 0209.40001
[22] A. G. Khovanskii: Fewnomials, Trans. AMS, 88, 1991. Zbl0728.12002 MR1108621 · Zbl 0728.12002
[23] I. Kupka: Abnormal extremals, Preprint, 1992.
[24] I. Kupka: Géométrie sous-Riemannienne, in Séminaire Bourbaki, 1996. MR1472545
[25] D.F. Lawden: Elliptic functions and applications, Springer-Verlag, New-York, 1989. Zbl0689.33001 MR1007595 · Zbl 0689.33001
[26] E. B. Lee, L. Markus: Foundations of optimal control theory, John Wiley and Sons, New-York, 1967. Zbl0159.13201 MR220537 · Zbl 0159.13201
[27] J. M. Lion, J. P. Rolin: Théorèmes de préparation pour les fonctions logarithmo-exponentielles, Annales de l’Institut Fourier, 47, 1997, 859-884. Zbl0873.32004 MR1465789 · Zbl 0873.32004 · doi:10.5802/aif.1583
[28] W. S. Liu and H. J. Susmann: Shortest paths for sub-Riemannian metries of rank two distributions, to appear in Trans. AMS. Zbl0843.53038 · Zbl 0843.53038
[29] S. Lojasiewicz, H. J. Sussmann: Some examples of reachable sets and optimal cost functions that fail to be subanalytic, SIAM J. Control and Optimization, 23, 1985, 584-598. Zbl0569.49029 MR791889 · Zbl 0569.49029 · doi:10.1137/0323037
[30] A. E. H. Love: A treatise of the mathematical theory of elasticity, Dover, 1944. Zbl0063.03651 MR10851 JFM47.0750.09 · Zbl 0063.03651
[31] S. B. Myers: Connections between differential geometry and topology, Duke Math. J., 1, 1935, 376-391. MR1545884 JFM61.0787.02 · Zbl 0012.27502 · doi:10.1215/S0012-7094-35-00126-0
[32] L. Pontriaguineet al.: Théorie mathématique des processus optimaux, Éditions MIR, Moscou, 1974. Zbl0289.49002 MR358482 · Zbl 0289.49002
[33] W. Respondek, M. Zhitomirskii: Feedback classification of nonlinear control Systems on 3-manifolds, to appear in Math. Control Systems and Signals. Zbl0925.93367 MR1403291 · Zbl 0925.93367
[34] M. Spivak: Differential geometry, Publish on Perish, Inc., Berkeley, 1979.
[35] R. Strichartz: Sub-Riemannian geometry, J. Differential geometry, 24, 1986, 221-263. Zbl0609.53021 MR862049 · Zbl 0609.53021
[36] J. Tannery, J. Molk: Eléments de la théorie des fonctions elliptiques, Tomes I à IV, Gauthier-Villars, Paris, 1896. JFM27.0335.01 · JFM 27.0335.01
[37] A. Weinstein: The cut-locus and conjugate-locus of a Riemannian manifold, Annals of Maths, 87, 1968, 29-41. Zbl0159.23902 MR221434 · Zbl 0159.23902 · doi:10.2307/1970592
[38] E. T. Whittaker, G. N. Watson: A course of modern analysis, Cambridge U. Press, New York, 1927. MR1424469 JFM53.0180.04 · JFM 53.0180.04
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