Wintgen ideal submanifolds: New examples, frame sequence and Möbius homogeneous classification

Abstract

In real space forms, any submanifold must satisfy the so-called DDVV inequality which relates the scalar curvature, the mean curvature, and the scalar normal curvature pointwise. When the equality is attained at each point, it is called a Wintgen ideal submanifold. This property is invariant under the conformal transformations. So we try to give a complete classification of this class of submanifolds. This is done under the additional assumption of Möbius homogeneity in this paper. Some new interesting examples are constructed using the real representation of SU(2), which turn out to constitute the majority of Möbius homogeneous Wintgen ideal submanifolds. The classification follows by constructing a frame sequence. This reveals a wonderful connection with the classical harmonic sequence of Riemann surfaces.