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 Möbius geometry. We classify Wintgen ideal submanfiolds of dimension m\ge \mathrm{3} and arbitrary codimension when a canonically defined 2-dimensional distribution {}_{\mathrm{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 {}_{\mathrm{2}} generates a k-dimensional integrable distribution {}_k<?Pub Caret?>and k<m, then similar reduction theorem holds true. This generalization when k = 3 has been proved in this paper.

Keywords

Wintgen ideal submanifold / DDVV inequality / super-conformal surface / super-minimal surface

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