$C\sp\infty$-regularity of a manifold as a function of its metric tensor.
Résumé
A basic theorem from differential geometry asserts that if the Riemann curvature tensor associated with a smooth field C of positive-definite symmetric matrices of order n vanishes in a simply-connected open subset Ω of R^n, then C is the metric tensor of a manifold isometrically immersed in R^n. If Ω is connected, then the isometric immersion Θ defined in this fashion is unique up to isometries of R^n. We prove that, if the set Ω is bounded and has a smooth boundary, then the mapping C → Θ is of class C^∞ between manifolds in appropriate Banach spaces.