A surface is a continuous function of its two fundamental forms

Philippe G. Ciarlet

doi:10.1016/S1631-073X(02)02538-4

A surface is a continuous function of its two fundamental forms
[Une surface est une fonction continue de ses deux formes fondamentales]

Philippe G. Ciarlet ^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

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

Résumé
Abstract

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.

Accepté le : 2002-08-26
Publié le : 2002-09-17

DOI : 10.1016/S1631-073X(02)02538-4

Affiliations des auteurs :

Philippe G. Ciarlet ^{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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Philippe G. Ciarlet. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02538-4/

Bibliographie
Cité par

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

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

[3] P.G. Ciarlet; F. Laurent 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] W. Klingenberg Eine Vorlesung über Differentialgeometrie, A Course in Differential Geometry, Springer-Verlag, Berlin, 1973 (English translation:, 1978, Springer-Verlag, Berlin)

Yuri Reshetnyak Estimates for completely integrable systems of differential operators and applications, Sobolev spaces in mathematics II. Applications in analysis and partial differential equations, New York, NY: Springer; Novosibirsk: Tamara Rozhkovskaya Publisher, 2009, pp. 311-327 | DOI:10.1007/978-0-387-85650-6_13 | Zbl:1172.35355
Philippe G. Ciarlet The continuity of a surface as a function of its two fundamental forms., Journal de Mathématiques Pures et Appliquées. Neuvième Série, Volume 82 (2003) no. 3, pp. 253-274 | DOI:10.1016/s0021-7824(03)00017-5 | Zbl:1042.53003

Cité par 2 documents. Sources : zbMATH

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