New compatibility conditions for the fundamental theorem of surface theory

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Original languageEnglish
Pages (from-to)273-278
Journal / PublicationComptes Rendus Mathematique
Issue number5
Online published21 Aug 2007
Publication statusPublished - 1 Sept 2007


The fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices (aαβ) of order two and a field of symmetric matrices (bαβ ) of order two together satisfy the Gauss and Codazzi-Mainardi equations in a connected and simply-connected open subset ω of ℝ2, then there exists an immersion θ : ω → ℝ3 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 ℝ3

In this Note, we identify new compatibility conditions, expressed again in terms of the functions aαβ and bαβ, that likewise lead to a similar existence and uniqueness theorem. These conditions take the form
1 A2 - ∂2 A1 + A1 A2 - A2 A1 = 0 in ω
where A1 and A2 are antisymmetric matrix fields of order three that are functions of the fields (aαβ) and (bαβ), the field (aαβ) appearing in particular through its square root. The unknown immersion θ : ω → ℝ3 is found in the present approach in function spaces 'with little regularity', viz., W2, p/loc (ω ; ℝ3), p > 2. To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 345 (2007).