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Article Dans Une Revue Journal de Mathématiques Pures et Appliquées Année : 2007

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

Philippe Ciarlet
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L. Gratie
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Cristinel Mardare
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C. Vallée
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Résumé

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.
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Dates et versions

hal-00117107 , version 1 (30-11-2006)

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Philippe Ciarlet, L. Gratie, O. Iosifescu, Cristinel Mardare, C. Vallée. Another approach to the fundamental theorem of Riemannian geometry in R3, by way of rotation fields. Journal de Mathématiques Pures et Appliquées, 2007, 87, pp.237-252. ⟨10.1016/j.matpur.2006.10.009⟩. ⟨hal-00117107⟩
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