Classification of Wintgen ideal surfaces in Euclidean 4-space with equal Gauss and normal curvatures

Abstract

Wintgen proved (C. R. Acad. Sci. Paris, 288:993–995, 1979) that the Gauss curvature K and the normal curvature KD of a surface in Euclidean 4-space \({\mathbb {E}^4}\) satisfy K + |KD| ≤ H2, where H2 is the squared mean curvature. A surface in \({\mathbb {E}^4}\) is called Wintgen ideal if it satisfies the equality case of the inequality identically. Wintgen ideal surfaces in \({\mathbb {E}^4}\) form an important family of surfaces, namely, surfaces with circular ellipse of curvature. In this article, we completely classify Wintgen ideal surfaces in \({\mathbb E^4}\) satisfying |K| = |KD| identically.