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

摘要

In 1992, C. Vallée showed that the metric tensor field C=ΘTΘ associated with a smooth enough immersion defined over an open set necessarily satisfies the compatibility relation where the matrix field Λ is defined in terms of the field U=C1/2 by 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 , then there exists, typically in spaces such as or , an immersion such that C=ΘTΘ in Ω. This global existence theorem thus provides an alternative to the fundamental theorem of Riemannian geometry for an open set in , 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=RC1/2 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=C1/2. 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.