TY - JOUR T1 - Two-dimensional almost-Riemannian structures with tangency points JF - Ann. Inst. H. Poincare Anal. Non Lineaire Y1 - 2010 A1 - Andrei A. Agrachev A1 - Ugo Boscain A1 - Grégoire Charlot A1 - Roberta Ghezzi A1 - Mario Sigalotti AB -
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.
PB - Elsevier VL - 27 UR - http://hdl.handle.net/1963/3870 U1 - 839 U2 - Mathematics U3 - Functional Analysis and Applications ER -