Another approach to the fundamental theorem of Riemannian geometry in R 3 , by way of rotation fields

• Published in: Journal de mathématiques pures et appliquées 2007-03, Vol.87 (3), p.237-252
• Main Authors: Ciarlet, Philippe G., Gratie, Liliana, Iosifescu, Oana, Mardare, Cristinel, Vallée, Claude
• Format: Article
• Language: English
• Subjects: Classical differential geometry; Fundamental theorem of Riemannian geometry; Nonlinear three-dimensional elasticity; Pfaff systems; Polar factorization
• ISSN: 0021-7824
• DOI: 10.1016/j.matpur.2006.10.009

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. En 1992, C. Vallée a montré que le champ C = ∇ Θ T ∇ Θ de tenseurs métriques associé à une immersion suffisamment régulière Θ : Ω → R 3 définie sur un ouvert Ω ⊂ R 3 vérifie nécessairement la relation de compatibilité CURL Λ + COF Λ = 0 in  Ω , où le champ Λ de matrices est défini en fonction du champ U = C 1 / 2 par Λ = 1 det U { U ( CURL U ) T U − 1 2 ( tr [ U ( CURL U ) T ] ) U } . L'objet principal de cet article est d'établir la réciproque suivante : Si un champ suffisamment régulier C de matrices symétriques définies positives d'ordre trois satisfait la relation de compatibilité ci-dessus dans un ouvert Ω ⊂ R 3 simplement connexe, alors il existe, typiquement dans des espaces tels que W loc 2 , ∞ ( Ω ; R 3 ) ou C 2 ( Ω ; R 3 ) , une immersion Θ : Ω → R 3 telle que C = ∇ Θ T ∇ Θ in Ω. Ce théorème d'existence global f