On embedded flat surfaces in \(S^3\)
There are many results on immersed flat surfaces \(M\) in the standard three-dimensional sphere \(S^3\) of constant sectional curvature one, some of which are reviewed (with slightly simplified proofs) in the first section of the present paper. However, the question of which of these surfaces are embedded was not addressed in these earlier results. This is the main concern of the authors in the present paper and they prove, among other results, that every embedded flat torus \(M \subset S^3\) has an antipodal symmetry and double covers an embedded torus in \({\mathbb{R}}P^3\). They also show how to construct every embedded flat torus \(M \subset S^3\), or \({\mathbb{R}}P^3\), from a pair of curves in the 2-sphere \(S^2\) satisfying a suitable condition. Finally, they show that there are embedded flat cylinders in \(S^3\), but they conjecture that it is impossible to embed \({\mathbb{R}}^2\) isometrically into \(S^3\).
scientific article; zbMATH DE number 1192503