New compatibility conditions for the fundamental theorem of surface theory

Abstract

The fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices ( a α β ) of order two and a field of symmetric matrices ( b α β ) of order two together satisfy the Gauss and Codazzi–Mainardi equations in a connected and simply-connected open subset ω of R 2 , then there exists an immersion θ : ω → R 3 such that these fields are the first and second fundamental forms of the surface θ ( ω ) and this surface is unique up to proper isometries in R 3 . In this Note, we identify new compatibility conditions, expressed again in terms of the functions a α β and b α β , that likewise lead to a similar existence and uniqueness theorem. These conditions take the form ∂ 1 A 2 − ∂ 2 A 1 + A 1 A 2 − A 2 A 1 = 0 in ω , where A 1 and A 2 are antisymmetric matrix fields of order three that are functions of the fields ( a α β ) and ( b α β ) , the field ( a α β ) appearing in particular through its square root. The unknown immersion θ : ω → R 3 is found in the present approach in function spaces ‘with little regularity’, viz., W loc 2 , p ( ω ; R 3 ) , p > 2 . To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 345 (2007).