ON THE RECOVERY AND CONTINUITY OF A SUBMANIFOLD WITH BOUNDARY
The fundamental theorem of Riemannian geometry asserts that a connected and simply-connected Riemannian space ω of ℝp can be isometrically immersed into the Euclidean space ℝp+q if and only if there exist tensors satisfying the Gauss–Ricci–Codazzi equations, in which case these immersions are uniquely determined up to isometries in ℝp+q. In this fashion, we can define a mapping which associates with these prescribed tensors the reconstructed submanifold. The purpose of this paper is twofold: under a smoothness assumption on the boundary of ω, we first establish an analogous result for the existence and uniqueness of a submanifold "with boundary" and then show that the mapping constructed in this fashion is locally Lipschitz-continuous with respect to the topology of the Banach spaces [Formula: see text].