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Two-dimensional almost-Riemannian structures with tangency points

TitleTwo-dimensional almost-Riemannian structures with tangency points
Publication TypeJournal Article
Year of Publication2009
AuthorsAgrachev, A, Boscain, U, Sigalotti, M, Ghezzi, R, Charlot, G

Two-dimensional almost-Riemannian structures are generalized Riemannian\r\nstructures on surfaces for which a local orthonormal frame is given by a Lie\r\nbracket generating pair of vector ?elds that can become collinear. We study the\r\nrelation between the topological invariants of an almost-Riemannian structure\r\non a compact oriented surface and the rank-two vector bundle over the surface\r\nwhich de?nes the structure. We analyse the generic case including the presence\r\nof tangency points, i.e. points where two generators of the distribution and\r\ntheir Lie bracket are linearly dependent. The main result of the paper provides\r\na classi?cation of oriented almost-Riemannian structures on compact oriented\r\nsurfaces in terms of the Euler number of the vector bundle corresponding to the\r\nstructure. Moreover, we present a Gauss?Bonnet formula for almost-Riemannian\r\nstructures with tangency points.


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