Surfaces in Euclidean 3-space whose normal bundles are tangentially biharmonic

TL;DR

This paper characterizes surfaces in Euclidean 3-space whose normal bundles are tangentially biharmonic, showing they are either minimal, spherical, or cylindrical, thus classifying such geometric structures.

Contribution

It provides a complete classification of surfaces with tangentially biharmonic normal bundles in Euclidean 3-space, linking this property to well-known surface types.

Findings

Surfaces with tangentially biharmonic normal bundles are minimal, spherical, or cylindrical.

The characterization is an if and only if condition.

The result extends understanding of biharmonic properties in submanifold geometry.

Abstract

A submanifold is said to be tangentially biharmonic if the bitension field of the isometric immersion that defines the submanifold has vanishing tangential component. The purpose of this paper is to prove that a surface in Euclidean $3$-space has tangentially biharmonic normal bundle if and only if it is either minimal, a part of a round sphere, or a part of a circular cylinder.