Isometric embeddings of the square flat torus in ambient space (Q2871886)

The present book is concerned with isometric embeddings of a square flat torus in the three dimensional Euclidean space \(\mathbb{E}^{3}.\) The existence of such embeddings was proved by John Nash and Nicolaas Kuiper in the mid 50s and states that any flat torus admits a \(C^{1}\) isometric embedding into \(\mathbb{E}^{3}.\) The fundamental step of the Nash-Kuiper proof is to construct from a strictly short embedding \(f_{1}\), i.e., an embedding that strictly shortens distances, another embedding \(f_{2}\) which is still strictly short but closer to an isometry. To this end, curves are lengthened in the normal neighborhood of the embedding \(f_{1}\) by turning them into spirals (Nash) or into oscillating curves (Kuiper). This fundamental step is then iteratively repeated to obtain an isometric embedding in the limit. However, the geometry of these isometric embeddings could barely be conceived from the original papers. Actually, the Nash-Kuiper result has long appeared as a separate and isolated result in Riemannian geometry and thus as a curiousity. For instance, it seemed to have no relevance to other important results of differential topology discovered by many mathematicians at this period like the classification of immersions by Smale-Hirsch (1959).NEWLINENEWLINE In the present work the authors provide an explicit construction of isometric embeddings based on the Convex Integration Theory introduced by Mikhail Gromov in the 70s. More precisely, it was in the 70s that Gromov extracted the basic idea of the Nash-Kuiper fundamental step and converted it into a powerful tool to solve partial differential equations: the Convex Integration Theory. The main ingredient of this theory is a barycentric formula which captures the spirit of both the Nash spiraling and the Kuiper oscillating processes. So, as a concequence, Gromov recovered the Nash-Kuiper theorem and the Smale-Hirsch classification of immersions. Also, other important theorems of differential topology resulted as corollaries of Gromov's theory.NEWLINENEWLINE In Chapter 2 of the book, the Convex Integration process is presented clearly in the case of isometric embeddings and based on it, the authors give rigorously an accessible proof of Nash-Kuiper's theorem. Finally, the authors turn the construction into a computer implementation leading us to the visualisation of an isometrically embedded flat torus. So, the whole construction is accompanied by very nice pictures.