No abstract

It is well-known that the curvature tensor is an isometric invariant of C 2 Riemannian manifolds. This invariant is at the origin of the rigidity observed in Riemannian geometry. In the mid 1950s, Nash amazed the world mathematical community by showing that this rigidity breaks down in regularity C 1 . This unexpected flexibility has many paradoxical consequences, one of them is the existence of C 1 isometric embeddings of flat tori into Euclidean three-dimensional space. In the 1970s and 1980s, M. Gromov, revisiting Nash's results introduced convex integration theory offering a general framework to solve this type of geometric problems. In this research, we convert convex integration theory into an algorithm that produces isometric maps of flat tori. We provide an implementation of a convex integration process leading to images of an embedding of a flat torus. The resulting surface reveals a C 1 fractal structure: Although the tangent plane is defined everywhere, the normal vector exhibits a fractal behavior. Isometric embeddings of flat tori may thus appear as a geometric occurrence of a structure that is simultaneously C 1 and fractal. Beyond these results, our implementation demonstrates that convex integration, a theory still confined to specialists, can produce computationally tractable solutions of partial differential relations.A geometric torus is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle. The standard parametrization of a geometric torus maps horizontal and vertical lines of a unit square to latitudes and meridians of the image surface. This unit square can also be seen as a torus; the top line is abstractly identified with the bottom line and so are the left and right sides. Because of its local Euclidean geometry, it is called a square flat torus. The standard parametrization now appears as a map from a square flat torus into the three-dimensional space having as its image a geometric torus. Although natural, this map distorts the distances: The lengths of latitudes vary whereas the lengths of the corresponding horizontal lines on the square remain constant.It was a long-held belief that this defect could not be fixed. In other words, it was presumed that no isometric embedding of the square flat torus-a differentiable injective map that preserves distances-could exist into three-dimensional space. In the mid 1950s Nash (1) and Kuiper (2) amazed the world mathematical community by showing that such an embedding actually exists. However, their proof relies on an intricated construction that makes it difficult to analyze the properties of the isometric embedding. In particular, these atypical embeddings have never been visualized. One strong motivation for such a visualization is the unusual regularity of the embedding: A continuously differentiable map that cannot be enhanced to be twice continuously differentiable. As a consequence, the image surface is smooth enough to have a tangent plane everywhere, but not sufficient ...

The volume of a unit vector field V of the sphere S n (n odd) is the volume of its image V (S n ) in the unit tangent bundle. Unit Hopf vector fields, that is, unit vector fields that are tangent to the fibre of a Hopf fibration S n → CP n−1 2 , are well known to be critical for the volume functional. Moreover, Gluck and Ziller proved that these fields achieve the minimum of the volume if n = 3 and they opened the question of whether this result would be true for all odd dimensional spheres. It was shown to be inaccurate on spheres of radius one. Indeed, Pedersen exhibited smooth vector fields on the unit sphere with less volume than Hopf vector fields for a dimension greater than five. In this article, we consider the situation for any odd dimensional spheres, but not necessarily of radius one. We show that the stability of the Hopf field actually depends on radius, instability occurs precisely if and only if r > 1 √ n−4 . In particular, the Hopf field cannot be minimum in this range. On the contrary, for r small, a computation shows that the volume of vector fields built by Pedersen is greater than the volume of the Hopf one thus, in this case, the Hopf vector field remains a candidate to be a minimizer. We then study the asymptotic behaviour of the volume; for small r it is ruled by the first term of the Taylor expansion of the volume. We call this term the twisting of the vector field. The lower this term is, the lower the volume of the vector field is for small r. It turns out that unit Hopf vector fields are absolute minima of the twisting. This fact, together with the stability result, gives two positive arguments in favour of the Gluck and Ziller conjecture for small r.