On the intrinsic Deszcz symmetries and the extrinsic Chen character of Wintgen ideal submanifolds

Abstract

In this paper it is shown that all Wintgen ideal submanifolds in ambient real space forms are Chen submanifolds. It is also shown that the Wintgen ideal submanifolds of dimension $ >3 $ in real space forms do intrinsically enjoy some curvature symmetries in the sense of Deszcz of their Riemann--Christoffel curvature tensor, of their Ricci curvature tensor and of their Weyl conformal curvature tensor.

On Wintgen ideal surfaces

“…A similar inequality holds for surfaces in pseudo-Euclidean 4-space E 4 2 with neutral metric [7,9]. Following L. Verstraelen et al [14,26], we call a surface M in E 4 Wintgen ideal if it satisfies the equality case of Wintgen's inequality identically. Obviously, Wintgen ideal surfaces in E 4 are exactly superminimal surfaces.…”

The Geometry of Universal Natural Shapes

“…This allows for the extrinsic study of manifolds as submanifolds in a Euclidean ambient space. Extrinsic information relates to the shapes that submanifolds assume in a surrounding space [98] (ideal submanifolds then try to minimize the amount of tension received from this space), analogous to a biological phenotype.…”

Classification Theorems for Space-Like Surfaces in 4-Dimensional Indefinite Space Forms with Index 2

“…Following [6,9,11], we call a surface in R 4 2 (c) Wintgen ideal if it satisfies the equality case of (5.2) identically. Wintgen ideal surfaces in E 4 2 satisfying |K| = |K D | are classified by the first author in [7] (see [6] for the classification of Wintgen ideal surfaces in E 4 satisfying |K| = |K D |).…”