Rotation fields and the fundamental theorem of Riemannian geometry in $ {\mathbb{R}}^{3}$

Ciarlet, Philippe G.; Gratie, Liliana; Iosifescu, Oana; Mardare, Cristinel; Vallée, Claude

doi:10.1016/j.crma.2006.08.007

Differential Geometry/Mathematical Problems in Mechanics

Rotation fields and the fundamental theorem of Riemannian geometry in

R^{3}

[Champs de rotations et le théorème fondamental de la géométrie riemannienne dans]

Présenté par : Philippe G. Ciarlet

Ciarlet, Philippe G.¹ ; Gratie, Liliana ² ; Iosifescu, Oana ³ ; Mardare, Cristinel ⁴ ; Vallée, Claude ⁵

¹ Department of Mathematics, City University of Hong Kong, 83, Tat Chee Avenue, Kowloon, Hong Kong
² Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, 83, Tat Chee Avenue, Kowloon, Hong Kong
³ Départment de mathématiques, université de Montpellier II, place Eugène-Bataillon, 34095 Montpellier cedex 5, France
⁴ Laboratoire Jacques-Louis Lions, université Pierre et Marie Curie, 4, place Jussieu, 75005 Paris, France
⁵ Laboratoire de mécanique des solides, université de Poitiers, 86962 Futuroscope-Chasseneuil cedex, France

Comptes Rendus. Mathématique, Tome 343 (2006) no. 6, pp. 415-421.

Résumé
Abstract

$R^{3}$ . Soit Ω un ouvert simplement connexe de $R^{3}$ . On montre dans cette Note que, si un champ suffisamment régulier U de matrices symétriques définies positives d'ordre trois satisfait la relation de compatibilité (due à C. Vallée)

CURL Λ + COF Λ = 0 dans Ω,

où le champ Λ de matrices est défini en fonction du champ U par

Λ = \frac{1}{\det U} {U {(CURL U)}^{T} U - \frac{1}{2} (tr [U {(CURL U)}^{T}]) U},

alors il existe, typiquement dans des espaces tels que

W_{loc}^{2, \infty} (Ω; R^{3})

C^{2} (Ω; R^{3})

, une immersion

Θ : Ω \to R^{3}

telle que

U^{2} = {\nabla Θ}^{T} \nabla Θ

in Ω. Dans cette approche, on cherche à identifier directement la factorisation polaire

\nabla Θ = RU

du gradient de l'immersion inconnue Θ en une rotation R et une extension pure

U = C^{1 / 2}

Let Ω be a simply-connected open subset of $R^{3}$ . We show in this Note that, if a smooth enough field U of symmetric and positive-definite matrices of order three satisfies the compatibility relation (due to C. Vallée)

CURL Λ + COF Λ = 0 in Ω,

where the matrix field Λ is defined in terms of the field U by

Λ = \frac{1}{\det U} {U {(CURL U)}^{T} U - \frac{1}{2} (tr [U {(CURL U)}^{T}]) U},

then there exists, typically in spaces such as

W_{loc}^{2, \infty} (Ω; R^{3})

C^{2} (Ω; R^{3})

, an immersion

Θ : Ω \to R^{3}

such that

U^{2} = {\nabla Θ}^{T} \nabla Θ

in Ω. In this approach, one directly seeks the polar factorization

\nabla Θ = RU

of the gradient of the unknown immersion Θ in terms of a rotation R and a pure stretch U.

Reçu le : 2006-08-15
Publié le : 2006-09-15

DOI : 10.1016/j.crma.2006.08.007

Affiliations des auteurs :

Ciarlet, Philippe G. ¹ ; Gratie, Liliana ² ; Iosifescu, Oana ³ ; Mardare, Cristinel ⁴ ; Vallée, Claude ⁵

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     author = {Ciarlet, Philippe G. and Gratie, Liliana and Iosifescu, Oana and Mardare, Cristinel and Vall\'ee, Claude},
     title = {Rotation fields and the fundamental theorem of {Riemannian} geometry in $ {\mathbb{R}}^{3}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {415--421},
     publisher = {Elsevier},
     volume = {343},
     number = {6},
     year = {2006},
     doi = {10.1016/j.crma.2006.08.007},
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Ciarlet, Philippe G.; Gratie, Liliana; Iosifescu, Oana; Mardare, Cristinel; Vallée, Claude. Rotation fields and the fundamental theorem of Riemannian geometry in $ {\mathbb{R}}^{3}$. Comptes Rendus. Mathématique, Tome 343 (2006) no. 6, pp. 415-421. doi : 10.1016/j.crma.2006.08.007. http://www.numdam.org/articles/10.1016/j.crma.2006.08.007/

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