A surface is a continuous function of its two fundamental forms (Q1852826)

The fundamental theorem of surface theory in the Euclidean \(\mathbb{E}^3\) implies a map \(F\) of any pair of elements in the spaces \(C^2(\omega, \mathbb{S}^2_>)\) and \(C(\omega,\mathbb{S}^2)\) \((\omega\) connected and simply connected open in \(\mathbb{R}^2\), \(\mathbb{S}^2_>\) space of symmetric resp. positive definite symmetric \(2\times 2\) matrices), satisfiying the Gauss and Codazzi equations, into the space of immersions \(C^3(\omega,\mathbb{E}^3)\) with these two fundamental forms up to arbitrary motions in \(\mathbb{E}^3\). Here the natural question arises whether there exist ad hoc topologies on \(C^2(\omega,\mathbb{S}^2_>)\times C^2( \omega, \mathbb{S}^2)\) and on \(C^3(\omega, \mathbb{E}^3)/R\) \((R\) relation of congruence in \(\mathbb{E}^3)\) which make the map \(F\) continuous. The author provides an affirmative answer to this question by the reduction to the analogous problem for the map \({\mathcal F}\) of the space \(C^2(\Omega,\mathbb{S}^3_>)\) \((\Omega\) connected and simply connected open in \(\mathbb{E}^3\), \(\mathbb{S}^3_>\) space of positive definite symmetric \(3\times 3\) matrices) with vanishing Riemannian sectional curvature into the space of isometric submersions \(C^3(\Omega,\mathbb{E}^3)/R\) (up to motions). The maps \(F\) and \({\mathcal F}\) become related by considering the normal congruence of a surface.