Mathematical Problems in Mechanics
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.

Given two fields of positive definite symmetric, and symmetric, matrices defined over a simply-connected open subset ωR2, 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 R3 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.

Etant donné deux champs suffisamment réguliers définis dans un ouvert simplement connexe ωR2, 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 R3 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.

Accepted:
Published online:
DOI: 10.1016/j.crma.2008.09.002
Ciarlet, Philippe G. 1; Iosifescu, Oana 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
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Ciarlet, Philippe G.; Iosifescu, Oana. 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. http://www.numdam.org/articles/10.1016/j.crma.2008.09.002/

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