Biharmonic submanifolds of $\mathbb{C}P^n$

TL;DR

This paper investigates proper-biharmonic submanifolds in complex projective spaces, establishing relations with Hopf-tubes, and characterizes biharmonic curves through their geometric properties, contributing new examples and insights.

Contribution

It introduces new families of proper-biharmonic submanifolds in complex projective spaces and relates their properties to Hopf-tubes, advancing understanding of biharmonic geometry.

Findings

Established relation between bitension fields of submanifolds and Hopf-tubes.

Produced new examples of proper-biharmonic submanifolds in $\,\mathbb{C}P^n$.

Characterized proper-biharmonic curves via curvatures and complex torsions.

Abstract

We give some general results on proper-biharmonic submanifolds of a complex space form and, in particular, of the complex projective space. These results are mainly concerned with submanifolds with constant mean curvature or parallel mean curvature vector field. We find the relation between the bitension field of the inclusion of a submanifold $\bar{M}$ in $\mathbb{C}P^n$ and the bitension field of the inclusion of the corresponding Hopf-tube in $\mathbb{S}^{2n+1}$. Using this relation we produce new families of proper-biharmonic submanifolds of $\mathbb{C}P^n$. We study the geometry of biharmonic curves of $\mathbb{C}P^n$ and we characterize the proper-biharmonic curves in terms of their curvatures and complex torsions.