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Keywords:
normal curvature; mean curvature; asymptotic field; hyperbolic point; isoptic curve; Wintgen ideal surface
normal curvature; mean curvature; asymptotic field; hyperbolic point; isoptic curve; Wintgen ideal surface
Summary:
The aim of this paper is to explore the relations between some geometric invariants associated with surfaces immersed in the Euclidean four-space, by investigating the extrinsic and intrinsic geometries of our surfaces from a global point of view, as well as considering the curvature ellipses of a given surface and their associated isoptic curves. We establish an elegant formula which shows how the isoptic curves of the curvature ellipses of a given surface are related to the asymptotic directions on the surface, and derive a Wintgen type inequality which provides both a simple relationship between the main intrinsic and extrinsic invariants, and a natural and geometric characterization of the hyperbolic points, on the given surface. To indicate an application of our Wintgen type inequality to the theory of M$\ddot{\mbox {o}}$bius invariant Euclidean submanifolds, we conclude the paper with a novel geometric result on a remarkable family of surfaces known as Wintgen ideal surfaces.
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