Another approach to the fundamental theorem of Riemannian geometry in R³, by way of rotation fields

In 1992, C. Vallée showed that the metric tensor field C = ∇Θ^T∇Θ associated with a smooth enough immersion Θ : Ω → R³ defined over an open set Ω ⊂ R³ necessarily satisfies the compatibility relation CURL Λ + COF Λ = 0 in Ω, where the matrix field Λ is defined in terms of the field U = C^1/2 by Λ = 1/det U {U (CURL U)^TU - ½ (tr[U(CURL U)^T])U}. The main objective of this paper is to establish the following converse: If a smooth enough field C of symmetric and positive-definite matrices of order three satisfies the above compatibility relation over a simply-connected open set Ω ⊂ R³, then there exists, typically in spaces such as W^{2, ∞}_loc (Ω ; R³) or C² (Ω ; R³), an immersion Θ : Ω → R³ such that C = ∇Θ^T∇Θ in Ω. This global existence theorem thus provides an alternative to the fundamental theorem of Riemannian geometry for an open set in R³, where the compatibility relation classically expresses that the Riemann curvature tensor associated with the field C vanishes in Ω. The proof consists in first determining an orthogonal matrix field R defined over Ω, then in determining an immersion Θ such that ∇Θ = RC^1/2 in Ω, by successively solving two Pfaff systems. In addition to its novelty, this approach thus also possesses a more "geometrical" flavor than the classical one, as it directly seeks the polar factorization ∇Θ = RU of the immersion gradient in terms of a rotation R and a pure stretch U = C^1/2. This approach also constitutes a first step towards the analysis of models in nonlinear three-dimensional elasticity where the rotation field is considered as one of the primary unknowns.