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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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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
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