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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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References:
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