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Two-dimensional almost-Riemannian structures are generalized Riemannianstructures on surfaces for which a local orthonormal frame is given by a Liebracket generating pair of vector fields that can become collinear.Generically, the singular set is an embedded one dimensional manifold and thereare three type of points: Riemannian points where the two vector fields arelinearly independent, Grushin points where the two vector fields are collinearbut their Lie bracket is not and tangency points where the two vector fieldsand their Lie bracket are collinear and the missing direction is obtained withone more bracket. Generically tangency points are isolated. In this paper westudy the Laplace-Beltrami operator on such a structure. In the case of acompact orientable surface without tangency points, we prove that theLaplace-Beltrami operator is essentially self-adjoint and has discretespectrum. As a consequence a quantum particle in such a structure cannot crossthe singular set and the heat cannot flow through the singularity. This is aninteresting phenomenon since when approaching the singular set (i.e. where thevector fields become collinear), all Riemannian quantities explode, butgeodesics are still well defined and can cross the singular set withoutsingularities. This phenomenon appears also in sub-Riemannian structure whichare not equiregular i.e. in which the grow vector depends on the point. We showthis fact by analyzing the Martinet case.

The Laplace-Beltrami operator in almost-Riemannian Geometry