 
Summary: TwoDimensional AlmostRiemannian Structures with
Tangency Points
A.A. Agrachev
, U. Boscain
, G. Charlot
, R. Ghezzi
, M. Sigalotti§
August 17, 2009
Abstract
Twodimensional almostRiemannian 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 almostRiemannian structure on a compact oriented surface and the ranktwo 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
almostRiemannian structures on compact oriented surfaces in terms of the Euler number of the
vector bundle corresponding to the structure. Moreover, we present a GaussBonnet formula
for almostRiemannian structures with tangency points.
1 Introduction
