In this talk I consider a surface embedded in a 3D contact sub-Riemannian manifold. Such a surface inherits a field of direction (with norm) from the ambient space. This field of directions is singular at characteristic points (i.e., where the surface is tangent to the set of admissible directions). In this talk we will study when the normed field of directions permits to give to the surface the structure of metric space (of ``SNCF'' type). I will also study how to define the heat and the Schroedinger equation on such a structure and if the singular points are ``accessible'' or not. When the singular points are accessible we will study self-adjoint extensions with Kirchhoff like boundary conditions.