Rotation fields and the fundamental theorem of Riemannian geometry in ℝ³

Let Ω be a simply-connected open subset of ℝ³. We show in this Note 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/det U){U(CURL U)^TU - (1/2)(tr[U(CURL U)^T])U}, 

then there exists, typically in spaces such as W_loc^2,∞(Ω; ℝ³) or C²(Ω; ℝ³), an immersion Θ : Ω → ℝ³ such that U² = ∇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.