Up to isometries, a deformation is a continuous function of its metric tensor. (Q1854682)

For \(\Omega\) a connected simply connected open set in \(\mathbb{R}^3\) with \(C= (g_{ij})\) a positive definite symmetric matrix field of class \(C^2\) in \(\Omega\), and the Riemann curvature tensor \(R_{qijk}\) vanishing in \(\Omega\), there exists a \(C^3\) immersion \(\Theta\) into Euclidean space \(E^3\) such that \(C= \nabla\Theta^T\nabla\Theta\) in \(\Omega\). For two \(C^1\) immersions \(\Theta\), \(\widetilde\Theta\) with \(C=\widetilde C\) in \(\Omega\), \(\Theta\) and \(\widetilde\Theta\) are the same up to Euclidean motion in \(E^3\), i.e. \(\Theta\equiv \widetilde\Theta\text{\,mod\,}R\). The authors construct topologies on the space of symmetric matrix fields of class \(C^2\) in \(\Omega\) and on \(C^3(\Omega; E^3)/R\) such that \(g_{ij}\mapsto C\text{\,mod\,}R\) is a continuous map.