On almost-Riemannian surfaces
[Sur les surfaces presque riemanniennes]
Séminaire de théorie spectrale et géométrie, Tome 29 (2010-2011), pp. 15-49.

Une structure presque riemannienne sur une surface est une structure riemannienne généralisée où les repères orthonormaux locaux sont donnés par des paires de champs de vecteurs qui peuvent être parallèles, mais dont l’algèbre de Lie engendrée a dimension 2 en tout point. En presque tout point de la surface, la distribution engendrée localement par ces repères a rang maximal, mais en général il existe un lieu, génériquement une courbe lisse, où la distribution a rang 1. Dans cet article on fournit une courte introduction à la géométrie presque-riemannienne de dimension 2, en soulignant les phénomènes nouveaux par rapport à la géométrie riemannienne. On présente quelques résultats décrivant des aspect topologiques, métriques et géométriques des surfaces presque riemanniennes d’un point de vue local et global.

An almost-Riemannian structure on a surface is a generalized Riemannian structure whose local orthonormal frames are given by Lie bracket generating pairs of vector fields that can become collinear. The distribution generated locally by orthonormal frames has maximal rank at almost every point of the surface, but in general it has rank 1 on a nonempty set which is generically a smooth curve. In this paper we provide a short introduction to 2-dimensional almost-Riemannian geometry highlighting its novelties with respect to Riemannian geometry. We present some results that investigate topological, metric and geometric aspects of almost-Riemannian surfaces from a local and global point of view.

DOI : 10.5802/tsg.284
Classification : 49J15, 53CXX, 34K35
Mots clés : almost-Riemannian geometry, geodesics, Grushin plane, Lipschitz classification, Pontryagin maximum principle, Gauss-Bonnet formula.
Ghezzi, Roberta 1

1 Department of Mathematical Sciences and Centre of Computational and Integrative Biology, Rutgers University Camden - Camden, 311 N 5th Street, Camden, NJ 08102, USA.
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Ghezzi, Roberta. On almost-Riemannian surfaces. Séminaire de théorie spectrale et géométrie, Tome 29 (2010-2011), pp. 15-49. doi : 10.5802/tsg.284. http://www.numdam.org/articles/10.5802/tsg.284/

[1] Agrachev, A. Compactness for sub-Riemannian length-minimizers and subanalyticity, Rend. Sem. Mat. Univ. Politec. Torino, Volume 56 (1998) no. 4, p. 1-12 (2001) Control theory and its applications (Grado, 1998) | MR | Zbl

[2] Agrachev, A.; Bonnard, B.; Chyba, M.; Kupka, I. Sub-Riemannian sphere in Martinet flat case, ESAIM Control Optim. Calc. Var., Volume 2 (1997), pp. 377-448 | Numdam | MR | Zbl

[3] Agrachev, A. A. A “Gauss-Bonnet formula” for contact sub-Riemannian manifolds, Dokl. Akad. Nauk, Volume 381 (2001) no. 5, pp. 583-585 | MR | Zbl

[4] Agrachev, A. A.; Barilari, D.; Boscain, U. Introduction to Riemannian and sub-Riemannian geometry (Lecture Notes), http://people.sissa.it/agrachev/agrachev_files/notes.html

[5] Agrachev, A. A.; Boscain, U.; Charlot, G.; Ghezzi, R.; Sigalotti, M. Two-Dimensional Almost-Riemannian Structures With Tangency Points, Proceedings of the 48th IEEE Conference on Decision and Control, December 16-18, 2009. Shangai, China.

[6] Agrachev, A. A.; Boscain, U.; Charlot, G.; Ghezzi, R.; Sigalotti, M. Two-dimensional almost-Riemannian structures with tangency points, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 27 (2010) no. 3, pp. 793-807 | DOI | Numdam | MR | Zbl

[7] Agrachev, Andrei; Boscain, Ugo; Sigalotti, Mario A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds, Discrete Contin. Dyn. Syst., Volume 20 (2008) no. 4, pp. 801-822 | MR | Zbl

[8] Agrachev, Andrei; Zelenko, Igor On feedback classification of control-affine systems with one- and two-dimensional inputs, SIAM J. Control Optim., Volume 46 (2007) no. 4, p. 1431-1460 (electronic) | DOI | MR | Zbl

[9] Agrachev, Andrei A.; Sachkov, Yuri L. Control theory from the geometric viewpoint, Encyclopaedia of Mathematical Sciences, 87, Springer-Verlag, Berlin, 2004 (Control Theory and Optimization, II) | MR | Zbl

[10] Bellaïche, André The tangent space in sub-Riemannian geometry, Sub-Riemannian geometry (Progr. Math.), Volume 144, Birkhäuser, Basel, 1996, pp. 1-78 | MR | Zbl

[11] Bonnard, B.; Caillau, J.-B.; Sinclair, R.; Tanaka, M. Conjugate and cut loci of a two-sphere of revolution with application to optimal control, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 26 (2009) no. 4, pp. 1081-1098 | EuDML | Numdam | MR | Zbl

[12] Bonnard, Bernard; Caillau, Jean Baptiste Singular Metrics on the Two-Sphere in Space Mechanics (Preprint 2008, HAL, vol. 00319299, pp. 1-25)

[13] Bonnard, Bernard; Charlot, Grégoire; Ghezzi, Roberta; Janin, Gabriel The Sphere and the Cut Locus at a Tangency Point in Two-Dimensional Almost-Riemannian Geometry, J. Dynam. Control Systems, Volume 17 (2011) no. 1, pp. 141-161 | MR | Zbl

[14] Bonnard, Bernard; Chyba, Monique Méthodes géométriques et analytiques pour étudier l’application exponentielle, la sphère et le front d’onde en géométrie sous-riemannienne dans le cas Martinet, ESAIM Control Optim. Calc. Var., Volume 4 (1999), p. 245-334 (electronic) | DOI | EuDML | Numdam | MR | Zbl

[15] Boscain, U.; Charlot, G.; Ghezzi, R.; Sigalotti, M. Lipschitz Classification of Almost-Riemannian Distances on Compact Oriented Surfaces, Journal of Geometric Analysis, pp. 1-18 (10.1007/s12220-011-9262-4)

[16] Boscain, Ugo; Chambrion, Thomas; Charlot, Grégoire Nonisotropic 3-level quantum systems: complete solutions for minimum time and minimum energy, Discrete Contin. Dyn. Syst. Ser. B, Volume 5 (2005) no. 4, pp. 957-990 | MR | Zbl

[17] Boscain, Ugo; Charlot, G.; Ghezzi, R. Normal forms and invariants for 2-dimensional almost-Riemannian structures (Preprint 2011, hal-00512380 v2, arXiv:1008.5036)

[18] Boscain, Ugo; Charlot, Grégoire Resonance of minimizers for n-level quantum systems with an arbitrary cost, ESAIM Control Optim. Calc. Var., Volume 10 (2004) no. 4, p. 593-614 (electronic) | DOI | EuDML | Numdam | MR | Zbl

[19] Boscain, Ugo; Charlot, Grégoire; Gauthier, Jean-Paul; Guérin, Stéphane; Jauslin, Hans-Rudolf Optimal control in laser-induced population transfer for two- and three-level quantum systems, J. Math. Phys., Volume 43 (2002) no. 5, pp. 2107-2132 | MR | Zbl

[20] Boscain, Ugo; Laurent, Camille The Laplace–Beltrami operator in almost-Riemannian Geometry (Preprint 2011, arXiv:1105.4687)

[21] Boscain, Ugo; Sigalotti, Mario High-order angles in almost-Riemannian geometry, Actes de Séminaire de Théorie Spectrale et Géométrie. Vol. 24. Année 2005–2006 (Sémin. Théor. Spectr. Géom.), Volume 25, Univ. Grenoble I, 2008, pp. 41-54 | EuDML | Numdam | MR | Zbl

[22] Franchi, Bruno; Lanconelli, Ermanno Une métrique associée à une classe d’opérateurs elliptiques dégénérés, Rend. Sem. Mat. Univ. Politec. Torino (1983) no. Special Issue, p. 105-114 (1984) Conference on linear partial and pseudodifferential operators (Torino, 1982) | MR | Zbl

[23] Grušin, V. V. A certain class of hypoelliptic operators, Mat. Sb. (N.S.), Volume 83 (125) (1970), pp. 456-473 | MR

[24] Hirsch, Morris W. Differential topology, Graduate Texts in Mathematics, 33, Springer-Verlag, New York, 1994 (Corrected reprint of the 1976 original) | MR | Zbl

[25] Jean, Frédéric Uniform estimation of sub-Riemannian balls, J. Dynam. Control Systems, Volume 7 (2001) no. 4, pp. 473-500 | DOI | MR | Zbl

[26] Kulkarni, Ravindra Shripad Curvature and metric, Ann. of Math. (2), Volume 91 (1970), pp. 311-331 | MR | Zbl

[27] Pelletier, Fernand Quelques propriétés géométriques des variétés pseudo-riemanniennes singulières, Ann. Fac. Sci. Toulouse Math. (6), Volume 4 (1995) no. 1, pp. 87-199 | EuDML | Numdam | MR | Zbl

[28] Pelletier, Fernand; Valère Bouche, Liane The problem of geodesics, intrinsic derivation and the use of control theory in singular sub-Riemannian geometry, Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992) (Sémin. Congr.), Volume 1, Soc. Math. France, Paris, 1996, pp. 453-512 | MR | Zbl

[29] Pontryagin, L. S.; Boltyanskiĭ, V. G.; Gamkrelidze, R. V.; Mishchenko, E. F. The Mathematical Theory of Optimal Processes, “Nauka”, Moscow, 1983 | MR | Zbl

[30] Vendittelli, Marilena; Oriolo, Giuseppe; Jean, Frédéric; Laumond, Jean-Paul Nonhomogeneous nilpotent approximations for nonholonomic systems with singularities, IEEE Trans. Automat. Control, Volume 49 (2004) no. 2, pp. 261-266 | DOI | MR

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