Tangentially biharmonic Lagrangian H-umbilical submanifolds in complex space forms

TL;DR

This paper classifies tangentially biharmonic Lagrangian H-umbilical submanifolds within complex space forms, expanding understanding of their geometric properties and the role of the bitension field in their characterization.

Contribution

It introduces a classification of tangentially biharmonic Lagrangian H-umbilical submanifolds in complex space forms, linking the concept to the normal bundle of hyperspheres.

Findings

Normal bundle of a round hypersphere can be immersed as a tangentially biharmonic Lagrangian H-umbilical submanifold

Classification results for these submanifolds in complex space forms

Extension of the concept of tangentially biharmonic submanifolds in complex geometry

Abstract

The notion of Lagrangian $H$-umbilical submanifolds was introduced by B. Y. Chen in 1997, and these submanifolds have appeared in several important problems in the study of Lagrangian submanifolds from the Riemannian geometric point of view. Recently, the author introduced the notion of tangentially biharmonic submanifolds, which are defined as submanifolds such that the bitension field of the inclusion map has vanishing tangential component. The normal bundle of a round hypersphere in $\mathbb{R}^n$ can be immersed as a tangentially biharmonic Lagrangian $H$-umbilical submanifold in $\mathbb{C}^n$. Motivated by this fact, we classify tangentially biharmonic Lagrangian $H$-umbilical submanifolds in complex space forms.