Local properties of almost-Riemannian structures in dimension 3

Ugo Boscain (1, 2) , Grégoire Charlot (3) , Moussa Gaye (2) , Paolo Mason (4)

1 GECO - Geometric Control Design
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
3 IF - Institut Fourier
4 L2S - Laboratoire des signaux et systèmes

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.

Mots clés: Almost-Riemannian structure; Sub-Riemannian geometry; cut locus; hypoelliptic diffusion

Domaines: Optimisation et contrôle [math.OC]; Géométrie différentielle [math.DG]; Equations aux dérivées partielles [math.AP]

Ugo Boscain, Grégoire Charlot, Moussa Gaye, Paolo Mason. Local properties of almost-Riemannian structures in dimension 3. Discrete and Continuous Dynamical Systems - Series A, 2014, 35 (9). ⟨hal-01017378⟩