Self-adjoint operators on surfaces in RnRn

Our aim in this paper is to define principal and characteristic directions at points on a smooth 2-dimensional surface in the Euclidean space R4R4 in such a way that their equations together with that of the asymptotic directions behave in the same way as the triple formed by their counterpart on smooth surfaces in the Euclidean space R3R3. The definitions we propose are derived from a more general approach, namely an analysis of self-adjoint operators on 2-dimensional smooth surfaces in the Euclidean space RnRn.