Two-dimensional almost-Riemannian structures with tangency points
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 3, pp. 793-807.

Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. We study the relation between the topological invariants of an almost-Riemannian structure on a compact oriented surface and the rank-two vector bundle over the surface which defines the structure. We analyse the generic case including the presence of tangency points, i.e. points where two generators of the distribution and their Lie bracket are linearly dependent. The main result of the paper provides a classification of oriented almost-Riemannian structures on compact oriented surfaces in terms of the Euler number of the vector bundle corresponding to the structure. Moreover, we present a Gauss–Bonnet formula for almost-Riemannian structures with tangency points.

@article{AIHPC_2010__27_3_793_0,
author = {Agrachev, A.A. and Boscain, U. and Charlot, G. and Ghezzi, R. and Sigalotti, M.},
title = {Two-dimensional almost-Riemannian structures with tangency points},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {793--807},
publisher = {Elsevier},
volume = {27},
number = {3},
year = {2010},
doi = {10.1016/j.anihpc.2009.11.011},
zbl = {1192.53029},
mrnumber = {2629880},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2009.11.011/}
}
Agrachev, A.A.; Boscain, U.; Charlot, G.; Ghezzi, R.; Sigalotti, M. Two-dimensional almost-Riemannian structures with tangency points. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 3, pp. 793-807. doi : 10.1016/j.anihpc.2009.11.011. http://archive.numdam.org/articles/10.1016/j.anihpc.2009.11.011/

[1] A. Agrachev, U. Boscain, M. Sigalotti, A Gauss–Bonnet-like formula on two-dimensional almost-Riemannian manifolds, Discrete Contin. Dyn. Syst. 20 no. 4 (2008), 801-822 | MR 2379474 | Zbl 1198.49041

[2] A.A. Agrachëv, A “Gauss–Bonnet formula” for contact sub-Riemannian manifolds, Dokl. Akad. Nauk 381 no. 5 (2001), 583-585 | MR 1890409 | Zbl 1044.53021

[3] A.A. Agrachev, Y.L. Sachkov, Control Theory from the Geometric Viewpoint, Encyclopaedia Math. Sci. vol. 87, Springer-Verlag, Berlin (2004) | MR 2062547 | Zbl 1062.93001

[4] A. Bellaïche, The tangent space in sub-Riemannian geometry, Sub-Riemannian Geometry, Progr. Math. vol. 144, Birkhäuser, Basel (1996), 1-78 | MR 1421822 | Zbl 0862.53031

[5] B. Bonnard, J.-B. Caillau, R. Sinclair, M. Tanaka, Conjugate and cut loci of a two-sphere of revolution with application to optimal control, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 no. 4 (2009), 1081-1098 | EuDML 78880 | Numdam | MR 2542715 | Zbl 1184.53036

[6] U. Boscain, T. Chambrion, G. Charlot, Nonisotropic 3-level quantum systems: Complete solutions for minimum time and minimum energy, Discrete Contin. Dyn. Syst. Ser. B 5 no. 4 (2005), 957-990 | MR 2170218 | Zbl 1084.81083

[7] U. Boscain, G. Charlot, J.-P. Gauthier, S. Guérin, H.-R. Jauslin, Optimal control in laser-induced population transfer for two- and three-level quantum systems, J. Math. Phys. 43 no. 5 (2002), 2107-2132 | MR 1893663 | Zbl 1059.81195

[8] U. Boscain, B. Piccoli, A short introduction to optimal control, T. Sari (ed.), Contrôle Non Linéaire et Applications, Hermann, Paris (2005), 19-66

[9] U. Boscain, M. Sigalotti, High-order angles in almost-Riemannian geometry, Actes de Séminaire de Théorie Spectrale et Géométrie, vol. 24, Année 2005–2006, Sémin. Théor. Spectr. Géom. vol. 25, Univ. Grenoble I (2008), 41-54 | EuDML 11229 | Numdam | MR 2478807 | Zbl 1159.53320

[10] B. Franchi, E. Lanconelli, Une métrique associée à une classe d'opérateurs elliptiques dégénérés, Conference on Linear Partial and Pseudodifferential Operators Torino, 1982 Rend. Semin. Mat. Univ. Politec. Torino no. Special Issue (1984), 105-114 | MR 745979 | Zbl 0553.35033

[11] V.V. Grušin, A certain class of hypoelliptic operators, Mat. Sb. (N.S.) 83 no. 125 (1970), 456-473 | EuDML 70289 | MR 279436

[12] M.W. Hirsch, Differential Topology, Grad. Texts in Math. vol. 33, Springer-Verlag, New York (1994) | MR 1336822 | Zbl 0121.18004

[13] V. Jurdjevic, Geometric Control Theory, Cambridge Stud. Adv. Math. vol. 52, Cambridge University Press, Cambridge (1997) | MR 1425878 | Zbl 0940.93005

[14] B. Malgrange, Ideals of Differentiable Functions, Tata Inst. Fund. Res. Stud. Math. vol. 3, Tata Institute of Fundamental Research, Bombay (1967) | MR 212575

[15] R. Montgomery, A tour of subriemannian geometries, their geodesics and applications, Math. Surveys Monogr. vol. 91, American Mathematical Society, Providence, RI (2002) | MR 1867362 | Zbl 1044.53022

[16] F. Pelletier, Quelques propriétés géométriques des variétés pseudo-riemanniennes singulières, Ann. Fac. Sci. Toulouse Math. (6) 4 no. 1 (1995), 87-199 | EuDML 73347 | Numdam | MR 1344719

[17] F. Pelletier, L. Valère Bouche, 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. vol. 1, Soc. Math. France, Paris (1996), 453-512 | MR 1427768 | Zbl 0877.53029