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 Θ̇ denote the equivalence class formed by all such deformations, and let ℱ: C → Θ̇ denote the mapping defined in this fashion. We establish here that the mapping ℱ is continuous, for certain natural metrizable topologies.