The laplace-beltrami operator in almost-riemannian geometry

Abstract

We study the Laplace-Beltrami operator of generalized Riemannian structures on orientable surfaces for which a local orthonormal frame is given by a pair of vector fields that can become collinear. Under the assumption that the structure is 2-step Lie bracket generating, we prove that the Laplace-Beltrami operator is essentially self-adjoint and has discrete spectrum. As a consequence, a quantum particle cannot cross the singular set (i.e., the set where the vector fields become collinear) and the heat cannot flow through the singularity. This is an interesting phenomenon since when approaching the singular set all Riemannian quantities explode, but geodesics are still well defined and can cross the singular set without singularities. This phenomenon also appears in sub-Riemannian structures which are not equiregular, i.e., when the growth vector depends on the point. We show this fact by analyzing the Martinet case. © Association des Annales de ľinstitut Fourier, 2013, tous droits réservés.