Two-Dimensional Almost-Riemannian Structures with Tangency Points

Details Two-Dimensional Almost-Riemannian Structures with Tangency Points Two-Dimensional Almost-Riemannian Structures with Tangency Points

2009-08-18

arxiv.org/abs/0908.2564v1

Mathematics

Differential Geometry

Scientific paper

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 ?elds 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 de?nes 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 classi?cation 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.

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