Two-dimensional almost-Riemannian structures with tangency points
Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 3, p. 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},
     publisher = {Elsevier},
     volume = {27},
     number = {3},
     year = {2010},
     pages = {793-807},
     doi = {10.1016/j.anihpc.2009.11.011},
     zbl = {1192.53029},
     mrnumber = {2629880},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 27 (2010) no. 3, pp. 793-807. doi : 10.1016/j.anihpc.2009.11.011. http://www.numdam.org/item/AIHPC_2010__27_3_793_0/

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