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 โฆ of R n , then there exists an immersion ฮ โ C 3 (โฆ; R n ), uniquely determined up to isometries in R n , such that C is the metric tensor field of the manifold ฮ(โฆ), then isometrically immersed in R n . Let ฮ denote the equivalence class of ฮ modulo isometries in R n and let F : C โ ฮ denote the mapping determined in this fashion.
The first objective of this paper is to show that, if โฆ satisfies a certain "geodesic property" (in effect a mild regularity assumption on the boundary โโฆ of โฆ) and if the field C and its partial derivatives of order 2 have continuous extensions to โฆ, the extension of the field C remaining positive-definite on โฆ, then the immersion ฮ and its partial derivatives of order 3 also have continuous extensions to โฆ.
The second objective is to show that, under a slightly stronger regularity assumption on โโฆ, the above extension result combined with a fundamental theorem of Whitney leads to a stronger extension result: There exist a connected open subset โฆ of R n containing โฆ and a field C of positivedefinite symmetric matrices of class C 2 on โฆ such that C is an extension of C and the Riemann curvature tensor associated with C still vanishes in โฆ.
The third objective is to show that, if โฆ 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 (โฆ) for the continuous extensions of the symmetric matrix fields C, and C 3 (โฆ) for the continuous extensions of the immersions ฮ.