no. 3
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},
     mrnumber = {2629880},
     zbl = {1192.53029},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2009.11.011/}
}
TY  - JOUR
AU  - Agrachev, A.A.
AU  - Boscain, U.
AU  - Charlot, G.
AU  - Ghezzi, R.
AU  - Sigalotti, M.
TI  - Two-dimensional almost-Riemannian structures with tangency points
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2010
SP  - 793
EP  - 807
VL  - 27
IS  - 3
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2009.11.011/
DO  - 10.1016/j.anihpc.2009.11.011
LA  - en
ID  - AIHPC_2010__27_3_793_0
ER  - 
%0 Journal Article
%A Agrachev, A.A.
%A Boscain, U.
%A Charlot, G.
%A Ghezzi, R.
%A Sigalotti, M.
%T Two-dimensional almost-Riemannian structures with tangency points
%J Annales de l'I.H.P. Analyse non linéaire
%D 2010
%P 793-807
%V 27
%N 3
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpc.2009.11.011/
%R 10.1016/j.anihpc.2009.11.011
%G en
%F AIHPC_2010__27_3_793_0
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 | Zbl

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

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

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

[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 | Numdam | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | Numdam | MR | Zbl

[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 | Zbl

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

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

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

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

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

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

[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 | Zbl

Cité par Sources :