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 (γαβ ) and (ραβ ) of order two satisfy appropriate compatibility relations in ω, then (γαβ ) and (ραβ ) 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 (γαβ) and (ραβ) 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 γαβ and ραβ and their covariant derivatives.