Wintgen ideal submanifolds with a low-dimensional integrable distribution (I)

TL;DR

This paper classifies Wintgen ideal submanifolds with an integrable 2-dimensional distribution in space forms, revealing their geometric structures as cones, cylinders, or rotational submanifolds over super-minimal surfaces.

Contribution

It provides a classification of higher-dimensional Wintgen ideal submanifolds with integrable distributions using Möbius geometry, extending previous understanding of their geometric configurations.

Findings

Classified Wintgen ideal submanifolds with integrable 2D distribution for dimension > 2

Examples include cones, cylinders, and rotational submanifolds

Results apply to space forms like spheres, Euclidean, and hyperbolic spaces

Abstract

A submanifold in space forms satisfies the well-known DDVV inequality due to De Smet, Dillen, Verstraelen and Vrancken. The submanifold attaining equality in the DDVV inequality at every point is called Wintgen ideal submanifold. As conformal invariant objects, Wintgen ideal submanifolds are studied in this paper using the framework of M\"{o}bius geometry. We classify Wintgen ideal submanfiolds of dimension $m>2$ and arbitrary codimension when a canonically defined 2-dimensional distribution $\mathbb{D}$ is integrable. Such examples come from cones, cylinders, or rotational submanifolds over super-minimal surfaces in spheres, Euclidean spaces, or hyperbolic spaces, respectively.