In 1992, C. Vallée showed that the metric tensor field C = ∇ Θ T ∇ Θ associated with a smooth enough immersion Θ : Ω → R 3 defined over an open set Ω ⊂ R 3 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 ) T U − 1 2 ( 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 ⊂ R 3 , then there exists, typically in spaces such as W loc 2 , ∞ ( Ω ; R 3 ) or C 2 ( Ω ; R 3 ) , an immersion Θ : Ω → R 3 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 R 3 , 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 = R C 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.