Abstract
In 1992, C. Vallée showed that the metric tensor field C associated with a smooth enough immersion of an open set Ω ⊂ R3 into the three-dimensional Euclidean space necessarily satisfies a compatibility relation defined in terms of the square root of C. The main objective of this paper is to establish the following converse: if a smooth enough field of symmetric and positive-definite matrices C of order three satisfies the above compatibility relation over a simply-connected open set Ω ⊂ R3, then there exists an immersion of Ω into the three-dimensional Euclidean space such that C is the metric tensor field associated with this immersion. This global existence theorem thus provides an alternative to the fundamental theorem of Riemannian geometry for an open set Ω ⊂ R3, where the compatibility relation classically expresses that the Riemann curvature tensor associated with the field C vanishes in Ω. In addition to its novelty, this approach possesses a more "geometrical" flavor than the classical one, as it directly seeks the polar factorization of the immersion gradient in terms of a rotation and a pure stretch. 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.