Abstract
Let Ω be a simply-connected open subset of R^3. We show that, if a smooth enough field U of symmetric and positive-definite matrices of order three satisfies the compatibility relation (due to C. Vallée) CURL Λ+COF Λ=0 in Ω,where the matrix field Λ is defined in terms of the field U by Λ=(1/detU){U(CURL U)^T U−(1/2)(tr[U(CURL U)T])U},then there exists, typically in spaces such as W^{2,∞}_loc(Ω;R^3) or C^2(Ω;R^3), an immersion Θ:Ω→R^3 such that U^2=∇Θ^T∇Θ in Ω. In this approach, one directly seeks the polar factorization ∇Θ=RU of the gradient of the unknown immersion Θ in terms of a rotation R and a pure stretch U.