On the recovery of a surface with prescribed first and second fundamental forms

The fundamental theorem of surface theory asserts that, 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 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 fight on the analogies existing between these two fundamental theorems of differential geometry. (C) 2002 Editions scientifiques et medicales Elsevier SAS. All rights reserved.