Explicit reconstruction of a displacement field on a surface by means of its linearized change of metric and change of curvature tensors

Let ω be a simply-connected open subset in R² and let θ : ω → R³ be a smooth immersion. If two symmetric matrix fields (γ_{α β}) and (ρ_{α β}) of order two satisfy appropriate compatibility relations in ω, then (γ_{α β}) and (ρ_{α β}) are the linearized change of metric and change of curvature tensor fields corresponding to a displacement vector field η of the surface θ (ω). We show here that, when the fields (γ_{α β}) and (ρ_{α β}) are smooth, the displacement vector η (y) at any point θ (y), y ∈ ω, of the surface θ (ω) can be explicitly computed by means of a "Cesàro-Volterra path integral formula on a surface", i.e., a path integral inside ω with endpoint y, and whose integrand is an explicit function of the functions γ_{α β} and ρ_{α β} and their covariant derivatives. To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008). © 2008 Académie des sciences.