The fundamental theorem of surface theory asserts that, if a field of positive definite symmetric matrices of order two and a<space>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 there exists a surface in R 3 with these fields as its first and second fundamental forms (global existence theorem) and this surface is unique up to isometries in R 3 (rigidity theorem).The aim of this paper is to provide a self-contained and essentially elementary proof of this theorem by showing how it can be established as a simple corollary of another well-known theorem of differential geometry, which asserts that, if the Riemann-Christoffel tensor associated with a field of positive definite symmetric matrices of order three vanishes in a connected and simply connected open subset of R 3 , then this field is the metric tensor field of an open set that can be isometrically imbedded in R 3 (global existence theorem) and this open set is unique up to isometries in R 3 (rigidity theorem). For convenience, we also give a self-contained proof of this theorem, as such a proof does not seem to be easy to locate in the existing literature.In addition to the simplicity of its principle, this approach has the merit to shed light on the analogies existing between these two fundamental theorems of differential geometry.
Le theoreme fondamental de la theorie des surfaces affirme que, si un champ de matrices symetriques definies positives d'ordre deux et un champ de matrices symetriques d'ordre deux verifient ensemble les equations de Gauss et de Codazzi-Mainardi dans un ouvert connexe et simplement connexe de<space>R 2 , alors il existe une surface dans R 3 dont ces champs sont les premiere et deuxieme formes fondamentales (theoreme d'existence globale) et cette surface est unique aux isometries de<space>R 3 pres (theoreme de rigidite).Le but de cet article est de donner une preuve autosuffisante et essentiellement elementaire de ce theoreme en montrant comment il peut etre obtenu comme un simple corollaire d'un autre theoreme bien connu de geometrie differentielle, qui affirme que, si le tenseur de Riemann-Christoffel associe a un champ de matrices symetriques definies positives d'ordre trois s'annule sur un ouvert connexe et simplement connexe de R 3 , alors ce champ est le tenseur metrique d'un ouvert qui peut etre plonge isometriquement dans R 3 (theoreme d'existence globale) et cet ouvert est unique aux isometries de R 3 pres (theoreme de rigidite). Par commodite, on donne egalement une preuve autosuffisante de ce theoreme, car une telle preuve ne semble pas aisee a localiser dans la litterature.Outre la simplicite de son principe, cette approche a le merite d'illustrer les analogies existant entre ces deux theoremes fondamentaux de la geometrie differentielle.