Two-Dimensional Almost-Riemannian Structures with Tangency Points

TL;DR

This paper classifies two-dimensional almost-Riemannian structures on compact surfaces, including tangency points, using topological invariants and introduces a Gauss-Bonnet formula for these structures.

Contribution

It provides a classification of almost-Riemannian structures with tangency points based on the Euler number of associated vector bundles and establishes a Gauss-Bonnet formula for them.

Findings

Classification of structures via Euler number

Analysis of tangency points in almost-Riemannian geometry

Gauss-Bonnet formula for structures with tangency points

Abstract

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.