Cesàro-volterra path integral formula on a surface

If a symmetric matrix field e = (e_ij) of order three satisfies the Saint-Venant compatibility 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 = ∇vT + ∇v in Ω. A classical result, due to Cesàro 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 e_ij 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 (γ _αβ) and (ρ_αβ) of order two satisfy appropriate compatibility relations in ω, then (γ_αβ) and (ρ_αβ) 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 "Cesàro-Volterra path integral formula on a surface" likewise holds when the fields (γ_αβ) and (ρ_αβ) 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 γ_αβ and ρ_αβ and their covariant derivatives. Such a formula has potential applications to the mathematical analysis and numerical simulation of linear "intrinsic" shell models. © 2009 World Scientific Publishing Company.