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Recovery of a manifold with boundary and its continuity as a function of its metric tensor. (English) Zbl 1088.74014
Summary: A basic theorem from differential geometry asserts that, if the Riemann curvature tensor associated with a field \({\mathbf C}\) of class \({\mathcal C}^2\) of positive-definite symmetric matrices of order \(n\) vanishes in a connected and simply-connected open subset \(\Omega\) of \(\mathbb{R}^n\) then there exists an immersion \(\Theta\in{\mathcal C}^3(\Omega;\mathbb{R}^n)\), uniquely determined up to isometries in \(\mathbb{R}^n\), such that \({\mathbf C}\) is the metric tensor field of the manifold \(\Theta(\Omega)\), then isometrically immersed in \(\mathbb{R}^n\). Let \(\dot \Theta\) denote the equivalence class of \(\Theta\) modulo isometries in \(\mathbb{R}^n\) and let \({\mathcal F}:{\mathbf C}\to\dot\Theta\) denote the mapping determined in this fashion.
The first objective of this paper is to show that, if \(\Omega\) satisfies a certain “geodesic property” (in effect a mild regularity assumption on the boundary \(\partial\Omega\) of \(\Omega)\) and if the field \({\mathbf C}\) and its partial derivatives of order \(\leq 2\) have continuous extensions to \(\overline \Omega\), the extension of the field \({\mathbf C}\) remaining positive-definite on \(\overline\Omega\), then the immersion \(\Theta\) and its partial derivatives of order \(\leq 3\) also have continuous extensions to \(\overline \Omega\).
The second objective is to show that, under a slightly stronger regularity assumption on \(\partial\Omega\), the above extension result combined with a fundamental theorem of Whitney leads to a stronger extension result: There exist a connected open subset \(\widetilde\Omega\) of \(\mathbb{R}^n\) containing \(\overline \Omega\) and a field \(\widetilde{\mathbf C}\) of positive-definite symmetric matrices of class \({\mathcal C}^2\) on \(\widetilde\Omega\) such that \(\widetilde{\mathbf C}\) is an extension of \(\mathbb{C}\) and the Riemann curvature tensor associated with \(\widetilde{\mathbf C}\) still vanishes in \(\widetilde \Omega\).
The third objective is to show that, if \(\Omega\) satisfies the geodesic property and is bounded, the mapping \({\mathcal F}\) can be extended to a mapping that is locally Lipschitz-continuous with respect to the topologies of the Banach spaces \({\mathcal C}^2 (\overline\Omega)\) for the continuous extensions of the symmetric matrix fields \({\mathbf C}\), and \({\mathcal C}^3(\overline\Omega)\) for the continuous extensions of the immersions \(\Theta\).

MSC:
74B20 Nonlinear elasticity
74A05 Kinematics of deformation
53Z05 Applications of differential geometry to physics
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