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Abstract

A basic theorem from differential geometry asserts that, if the Riemann curvature tensor associated with a field C of class C 2 of positive-definite symmetric matrices of order n vanishes in a connected and simply-connected open subset Ω of R n, then there exists an immersion Θ ∈ C 3 (Ω; R n), uniquely determined up to isometries in R n, such that C is the metric tensor field of the manifold Θ(Ω), then isometrically immersed in R n. Let ˙ Θ denote the equivalence class of Θ modulo isometries in R n and let F: C → ˙ Θ denote the mapping determined in this fashion. The first objective of this paper is to show that, if Ω satisfies a certain “geodesic property ” (in effect a mild regularity assumption on the boundary ∂Ω of Ω) and if the field C and its partial derivatives of order ≤ 2 have continuous extensions to Ω, the extension of the field C remaining positive-definite on Ω, then the immersion Θ and its partial derivatives of order ≤ 3 also have continuous extensions to Ω. The second objective is to show that, under a slightly stronger regularity assumption on ∂Ω, the above extension result combined with a fundamental theorem o