A surface is a continuous function of its two fundamental forms
Comptes Rendus. Mathématique, Volume 335 (2002) no. 7, pp. 609-614.

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 2 , then these fields are the first and second fundamental forms of a surface in 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.

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 2 , alors ces champs sont les première et deuxième formes fondamentales d'une surface dans 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.

Accepted:
Published online:
DOI: 10.1016/S1631-073X(02)02538-4
Ciarlet, Philippe G. 1, 2

1 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4, place Jussieu, 75005 Paris, France
2 Department of Mathematics, City University of Hong Kong, 83, Tat Chee avenue, Kowloon, Hong Kong
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Ciarlet, Philippe G. A surface is a continuous function of its two fundamental forms. Comptes Rendus. Mathématique, Volume 335 (2002) no. 7, pp. 609-614. doi : 10.1016/S1631-073X(02)02538-4. http://www.numdam.org/articles/10.1016/S1631-073X(02)02538-4/

[1] P.G. Ciarlet, On the continuity of a surface as a function of its two fundamental forms, to appear

[2] Ciarlet, P.G.; Larsonneur, F. On the recovery of a surface with prescribed first and second fundamental forms, J. Math. Pures Appl., Volume 81 (2002), pp. 167-185

[3] Ciarlet, P.G.; Laurent, F. Up to isometries, a deformation is a continuous function of its metric tensor, C. R. Acad. Sci. Paris, Série I, Volume 335 (2002), pp. 489-493

[4] P.G. Ciarlet, F. Laurent, On the continuity of a deformation as a function of its Cauchy–Green tensor, 2002, to appear

[5] Klingenberg, W. Eine Vorlesung über Differentialgeometrie, A Course in Differential Geometry, Springer-Verlag, Berlin, 1973 (English translation:, 1978, Springer-Verlag, Berlin)

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