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

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Original languageEnglish
Pages (from-to)237-252
Journal / PublicationJournal des Mathematiques Pures et Appliquees
Issue number3
Online published1 Dec 2006
Publication statusPublished - Mar 2007


In 1992, C. Vallée showed that the metric tensor field C = ∇ΘT∇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 = C1/2 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 W2, ∞loc (Ω ; R3) or C2 (Ω ; R3), an immersion Θ : Ω → R3 such that C = ∇ΘT∇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 = 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.

Research Area(s)

  • Classical differential geometry, Fundamental theorem of Riemannian geometry, Nonlinear three-dimensional elasticity, Pfaff systems, Polar factorization

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