Recovery of a displacement field on a surface from its linearized change of metric and change of curvature tensors

We establish that the components of the linearized change of metric and change of curvature tensors associated with a displacement field of a surface in ℝ³ must satisfy compatibility conditions, which are the analogues 'on a surface' of the Saint Venant equations in three-dimensional elasticity. 
We next show that, conversely, if two symmetric matrix fields of order two satisfy these compatibility conditions over a simply-connected surface S ⊂ ℝ³, then they are the linearized change of metric and change of curvature tensors associated with a displacement field of the surface S. To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).