Recovery of a manifold with boundary and its continuity as a function of its metric tensor

AbstractA basic theorem from differential geometry asserts that, if the Riemann curvature tensor associated with a field C of class C2 of positive-definite symmetric matrices of order n vanishes in a connected and simply-connected open subset Ω of Rn, then there exists an immersion Θ∈C3(Ω;Rn), uniquely determined up to isometries in Rn, such that C is the metric tensor field of the manifold Θ(Ω), then isometrically immersed in Rn. Let Θ̇ denote the equivalence class of Θ modulo isometries in Rn and let F:C→Θ̇ 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 Rn containing Ω and a field C of positive-definite symmetric matrices of class C2 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 C2(Ω) for the continuous extensions of the symmetric matrix fields C, and C3(Ω) for the continuous extensions of the immersions Θ