A new approach to the fundamental theorem of surface theory, by means of the Darboux–Vallée–Fortuné compatibility relation

Abstract

Let ω be a simply-connected open subset of R 2 . Given two smooth enough fields of positive definite symmetric, and symmetric, matrices defined over ω, 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 first and second fundamental forms of the surface θ ( ω ) . We revisit here this classical result by establishing 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 . 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.