Wintgen ideal submanifolds: reduction theorems and a coarse classification
Wintgen ideal submanifolds in space forms are those ones attaining equality pointwise in the so-called DDVV inequality \(s\leq c+|H|^2 -s_N\), see [Zbl 1054.53075; Zbl 0515.53044; Zbl 0421.53003], which relates the scalar curvature, the mean curvature and the scalar normal curvature. Those submanifolds attaining the equality at every point are called \textit{Wintgen ideal submanifolds}. As conformal invariant objects, they are suitable to study in the framework of Möbius geometry. This paper continues authors's previous work [Zbl 1335.53015] in this program, showing that Wintgen ideal submanifolds can be divided into three classes: the reducible ones, the irreducible minimal ones in space forms (up to Möbius transformations), and the generic (irreducible) ones. The reducible Wintgen ideal submanifolds have a specific low-dimensional integrable distribution, which allows the authors to get the most general reduction theorem, saying that they are Möbius equivalent to cones, cylinders, or rotational surfaces generated by minimal Wintgen ideal submanifolds in lower-dimensional space forms.
scientific article; zbMATH DE number 6869153
scientific article