Normal forms and invariants for 2-dimensional almost-Riemannian structures

TL;DR

This paper develops normal forms and invariants for 2D almost-Riemannian structures, classifying points and constructing canonical curves to recognize local isometries, with detailed analysis of various point types and their geometric properties.

Contribution

It introduces a comprehensive method for finding normal forms and complete invariants at different point types in 2D almost-Riemannian structures, including tangency points.

Findings

Normal forms are established for Riemannian, Grushin, and tangency points.

Canonical curves are constructed for each point type to recognize local isometries.

The cut locus analysis shows non-smoothness, leading to the use of curvature crests and valleys for invariants.

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. Generically, there are three types of points: Riemannian points where the two vector fields are linearly independent, Grushin points where the two vector fields are collinear but their Lie bracket is not, and tangency points where the two vector fields and their Lie bracket are collinear and the missing direction is obtained with one more bracket. In this paper we consider the problem of finding normal forms and functional invariants at each type of point. We also require that functional invariants are "complete" in the sense that they permit to recognize locally isometric structures. The problem happens to be equivalent to the one of finding a smooth canonical parameterized curve passing through the point and being transversal to the distribution. For Riemannian points such that the gradient of the Gaussian curvature $K$ is different from zero, we use the level set of $K$ as support of the parameterized curve. For Riemannian points such that the gradient of the curvature vanishes (and under additional generic conditions), we use a curve which is found by looking for crests and valleys of the curvature. For Grushin points we use the set where the vector fields are parallel. Tangency points are the most complicated to deal with. The cut locus from the tangency point is not a good candidate as canonical parameterized curve since it is known to be non-smooth. Thus, we analyse the cut locus from the singular set and we prove that it is not smooth either. A good candidate appears to be a curve which is found by looking for crests and valleys of the Gaussian curvature. We prove that the support of such a curve is uniquely determined and has a canonical parametrization.