Continuity of a Deformation as a Function of its Cauchy-Green Tensor

If the Riemann-Christoffel tensor associated with a field C of class 𝒞² of positive-definite symmetric matrices of order 3 vanishes in a simply connected open subset Ω⊂ℝ³, then this field is the Cauchy-Green tensor field associated with a deformation Θ of class 𝒞³ of the set Ω, and Θ is uniquely determined up to isometries of ℝ³. Let denote the equivalence class formed by all such deformations, and let denote the mapping defined in this fashion. We establish here that the mapping ℱ is continuous, for certain natural metrizable topologies.