Local properties of almost-Riemannian structures in dimension 3

TL;DR

This paper investigates the local geometric and analytic properties of 3D almost-Riemannian manifolds, focusing on singularities, abnormal extremals, and heat diffusion, extending the well-studied 2D case to higher dimensions.

Contribution

It introduces the study of 3D almost-Riemannian structures, analyzing singularities, abnormal extremals, and local normal forms, which were not previously explored in depth.

Findings

Classification of generic singularities

Construction of local normal forms

Preliminary results on heat diffusion

Abstract

A 3D almost-Riemannian manifold is a generalized Riemannian manifold defined locally by 3 vector fields that play the role of an orthonormal frame, but could become collinear on some set $\Zz$ called the singular set. Under the Hormander condition, a 3D almost-Riemannian structure still has a metric space structure, whose topology is compatible with the original topology of the manifold. Almost-Riemannian manifolds were deeply studied in dimension 2. In this paper we start the study of the 3D case which appear to be reacher with respect to the 2D case, due to the presence of abnormal extremals which define a field of directions on the singular set. We study the type of singularities of the metric that could appear generically, we construct local normal forms and we study abnormal extremals. We then study the nilpotent approximation and the structure of the corresponding small spheres. We finally give some preliminary results about heat diffusion on such manifolds.