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 of Whitney leads to a stronger extension result: There exist a connected open subset Ω' of R^n containing Ω and a field C' of positive-definite symmetric matrices of class C^2 on Ω' such that C' is an extension of C and the Riemann curvature tensor associated with C' still vanishes in Ω'. The third objective is to show that, if Ω satisfies the geodesic property and is bounded, the mapping F can be extended to a mapping that is locally Lipschitz-continuous with respect to the topologies of the Banach spaces C^2(closure(Ω)) for the continuous extensions of the symmetric matrix fields C, and C^3(closure(Ω)) for the continuous extensions of the immersions Θ.