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Two-Dimensional Almost-Riemannian Structures with Tangency Points

Summary: Two-Dimensional Almost-Riemannian Structures with
Tangency Points
A.A. Agrachev
, U. Boscain
, G. Charlot
, R. Ghezzi
, M. Sigalotti§
August 17, 2009
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 Gauss­Bonnet formula
for almost-Riemannian structures with tangency points.
1 Introduction


Source: Agrachev, Andrei - Functional Analysis Sector, Scuola Internazionale Superiore di Studi Avanzati (SISSA)


Collections: Engineering; Mathematics