A new approach to the fundamental theorem of surface theory, by means of the Darboux-Vallee-Fortune compatibility relation

Let omega be a simply-connected open subset of R-2. Given two smooth enough fields of positive definite symmetric, and symmetric, matrices defined over omega, the fundamental theorem of surface theory asserts that, if these fields satisfy the Gauss and Codazzi-Mainardi relations in omega, then there exists an immersion theta from omega into R-3 such that these fields are the first and second fundamental forms of the surface theta(omega). We revisit here this classical result by establishing that a new compatibility relation, shown to be necessary by C. Vallee and D. Fortune 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 theta. This approach also constitutes a first step toward the analysis of models for nonlinear elastic shells where the rotation field along the middle surface is considered as one of the primary unknowns. (C) 2009 Elsevier Masson SAS. All rights reserved.