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 R-3 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 subset of R-3, then they are the linearized change of metric and change of curvature tensors associated with a displacement field of the surface S.