Justification of the Darboux–Vallée–Fortuné compatibility relation in the theory of surfaces

Philippe G. Ciarlet; Oana Iosifescu

doi:10.1016/j.crma.2008.09.002

Mathematical Problems in Mechanics

Justification of the Darboux–Vallée–Fortuné compatibility relation in the theory of surfaces
[Justification de la condition de compatibilité de Darboux–Vallée–Fortuné en théorie des surfaces]

Présenté par : Philippe G. Ciarlet

Philippe G. Ciarlet ¹ ; Oana Iosifescu ²

¹ Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
² Département de mathématiques, Université de Montpellier II, place Eugène-Bataillon, 34095 Montpellier cedex 5, France

Comptes Rendus. Mathématique, Volume 346 (2008) no. 21-22, pp. 1197-1202.

Résumé
Abstract

Etant donné deux champs suffisamment réguliers définis dans un ouvert simplement connexe $ω \subset R^{2}$ , l'un de matrices symétriques définies positives et l'autre de matrices symétriques, le théorème fondamental de la théorie des surfaces affirme que, si ces deux champs satisfont les relations de Gauss et Codazzi–Mainardi dans ω, alors il existe une immersion θ de ω dans $R^{3}$ telle que ces champs soient les deux formes fondamentales de la surface $θ (ω)$ .

On montre ici qu'une nouvelle relation de compatibilité, dont C. Vallée et D. Fortuné ont montré en 1996 la nécessité en suivant une idée de G. Darboux, est également suffisante pour l'existence d'une telle immersion θ.

Given two fields of positive definite symmetric, and symmetric, matrices defined over a simply-connected open subset $ω \subset R^{2}$ , the fundamental theorem of surface theory asserts that, if these fields satisfy the Gauss and Codazzi–Mainardi relations in ω, then there exists an immersion θ from ω into $R^{3}$ such that these fields are the two fundamental forms of the surface $θ (ω)$

We show here that a new compatibility relation, shown to be necessary by C. Vallée and D. Fortuné in 1996 through the introduction, following an idea of G. Darboux, of a rotation field on a surface, is also sufficient for the existence of such an immersion θ.

Accepté le : 2008-08-05
Publié le : 2008-10-02

DOI : 10.1016/j.crma.2008.09.002

Affiliations des auteurs :

Philippe G. Ciarlet ¹ ; Oana Iosifescu ²

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     title = {Justification of the {Darboux{\textendash}Vall\'ee{\textendash}Fortun\'e} compatibility relation in the theory of surfaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1197--1202},
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     number = {21-22},
     year = {2008},
     doi = {10.1016/j.crma.2008.09.002},
     language = {en},
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Philippe G. Ciarlet; Oana Iosifescu. Justification of the Darboux–Vallée–Fortuné compatibility relation in the theory of surfaces. Comptes Rendus. Mathématique, Volume 346 (2008) no. 21-22, pp. 1197-1202. doi : 10.1016/j.crma.2008.09.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.09.002/

Bibliographie
Cité par

[1] P.G. Ciarlet; L. Gratie; O. Iosifescu; C. Mardare; C. Vallée Another approach to the fundamental theorem of Riemannian geometry in $R^{3}$ , by way of rotation fields, J. Math. Pures Appl., Volume 87 (2007), pp. 237-252

[2] P.G. Ciarlet, O. Iosifescu, A new approach to the fundamental theorem of surface theory, by means of the Darboux–Vallée–Fortuné compatibility relation, J. Mach. Pures Appl., in press

[3] G. Darboux Leçons sur la Théorie Générale des Surfaces et les Applications Géométriques du Calcul Infinitésimal, vols. 1–4, Gauthier-Villars, Paris, 1894 (re-published in 2000 by the American Mathematical Society, Providence, RI)

[4] C. Mardare On the recovery of a manifold with prescribed metric tensor, Anal. Appl., Volume 1 (2003), pp. 433-453

[5] S. Mardare On Pfaff systems with $L^{p}$ coefficients and their applications in differential geometry, J. Math. Pures Appl., Volume 84 (2005), pp. 1659-1692

[6] W. Pietraszkiewicz; M.L. Szwabowicz Determination of the midsurface of a deformed shell from prescribed fields of surface strains and bendings, Internat. J. Solids Structures, Volume 44 (2007), pp. 6163-6172

[7] W. Pietraszkiewicz, M.L. Szwabowicz, C. Vallée, Determination of the midsurface of a deformed shell from prescribed surface strains and bendings via the polar decomposition, Internat. J. Non-Linear Mech. (2008), in press

[8] W. Pietraszkiewicz; C. Vallée A method of shell theory in determination of the surface from components of its two fundamental forms, Z. Angew. Math. Mech., Volume 87 (2007), pp. 603-615

[9] R.T. Shield The rotation associated with large strains, SIAM J. Appl. Math., Volume 25 (1973), pp. 483-491

[10] C. Vallée Compatibility equations for large deformations, Internat. J. Engrg. Sci, Volume 30 (1992), pp. 1753-1757

[11] C. Vallée; D. Fortuné Compatibility equations in shell theory, Internat. J. Engrg. Sci., Volume 34 (1996), pp. 495-499

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