Flat torus in the ambient space and implementation of convex integration
Résumé
In the mid 1950's, J. Nash and N. Kuiper showed it was possible to isometrically embed a flat torus in three dimensional Euclidean space. This result was counter-intuitive since the Gauss curvature prevents those embeddings to be of class C^2. The maps of Nash and Kuiper are of class C^1, which means in particular the existence of a tangent space at each point of the embeddings. Based on a technic invented by M. Gromov, the convex integration theory, we have been able to build an isometric embedding of the square flat torus in the ambient space and to partly understand its paradoxal geometry