Cesàro-Volterra path integral formula on a surface

If a symmetric matrix field e = (eij ) of order three satisfies the Saint-Venant compati- bility relations in a simply-connected open subset Ω of R3, then e is the linearized strain tensor field of a displacement field v of Ω, i.e. e = 1 (∇vT + ∇v) in Ω. A classical 2 result, due to Ces`aro and Volterra, asserts that, if the field e is smooth, the unknown displacement field v(x) at any point x ∈ Ω can be explicitly written as a path integral inside Ω with endpoint x, and whose integrand is an explicit function of the functions eij and their derivatives. Now 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 field corresponding to a displacement vector field η of the surface θ(ω). We show here that a “Ces`aro–Volterra path integral formula on a surface” likewise holds when the fields (γαβ) and (ραβ) are smooth. This means that the displacement vector η(y) at any point θ(y), y ∈ ω, of the surface θ(ω) can be explicitly computed as 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.