Wintgen ideal submanifolds with a low-dimensional integrable distribution

Abstract

Submanifolds in space forms satisfy the well-known DDVV inequality. A submanifold attaining equality in this inequality pointwise is called a Wintgen ideal submanifold. As conformal invariant objects, Wintgen ideal submanifolds are investigated in this paper using the framework of Mobius geometry. We classify Wintgen ideal submanfiolds of dimension m ⩽ 3 and arbitrary codimension when a canonically defined 2-dimensional distribution \(\mathbb{D}_2\) is integrable. Such examples come from cones, cylinders, or rotational submanifolds over super-minimal surfaces in spheres, Euclidean spaces, or hyperbolic spaces, respectively. We conjecture that if \(\mathbb{D}_2\) generates a k-dimensional integrable distribution \(\mathbb{D}_k\) and k < m, then similar reduction theorem holds true. This generalization when k = 3 has been proved in this paper.