A surface is a continuous function of its two fundamental forms

Ciarlet, Philippe G.

Comptes Rendus. Mathématique, Tome 335 (2002) no. 7, pp. 609-614.

Si un champ de matrices symétriques définies positives d'ordre deux et un champ de matrices symétriques d'ordre deux vérifient ensemble les équations de Gauß et de Codazzi–Mainardi dans un ouvert connexe et simplement connexe de ${ℝ}^2$, alors ces champs sont les première et deuxième formes fondamentales d'une surface dans ${ℝ}^3$, unique aux isométries près. On établit ici qu'une surface définie de cette façon varie continûment en fonction de ses deux formes fondamentales, pour des topologies métrisables convenables.

If a field of positive definite symmetric matrices of order two and a field of symmetric matrices of order two together satisfy the Gauß and Codazzi–Mainardi equations in a connected and simply connected open subset of ${ℝ}^2$, then these fields are the first and second fundamental forms of a surface in ${ℝ}^3$, unique up to isometries. It is shown here that a surface defined in this fashion varies continuously as a function of its two fundamental forms, for ad hoc metrizable topologies.

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