Extension of a Riemannian metric with vanishing curvature.
The authors prove the following theorem: Let \(\Omega\) be a simply-connected open subset of \(\mathbb R^n\) satisfying the strong geodesic property and let be given a Riemannian metric \((g_{ij})\) of class \(\mathcal{C}^2\) and of vanishing curvature in \(\Omega\), such that the functions \(g_{ij}\) and their partial derivatives of order \(\leqslant 2\) have continuous extensions to \(\overline\Omega\). Then there exists a connected open subset \(\widetilde\Omega\) of \(\mathbb R^n\) containing \(\overline\Omega\) and a Riemannian metric \((\tilde g_{ij})\) of class \(\mathcal{C}^2\) and of vanishing curvature in \(\widetilde\Omega\) that extends the metric \((g_{ij})\).
scientific article; zbMATH DE number 2057120