Explicit formulas for biharmonic submanifolds in Sasakian space forms

TL;DR

This paper classifies biharmonic Legendre curves and constructs new examples of biharmonic submanifolds in spheres, especially in , using explicit formulas and transformations in Sasakian space forms.

Contribution

It provides explicit parametric equations for biharmonic Legendre curves and introduces a method to generate biharmonic submanifolds via Reeb flow in Sasakian space forms.

Findings

Explicit formulas for biharmonic Legendre curves in spheres

New examples of biharmonic submanifolds in  and spheres

Transformation method using Reeb flow for biharmonic submanifolds

Abstract

We classify the biharmonic Legendre curves in a Sasakian space form, and obtain their explicit parametric equations in the $(2n+1)$-dimensional unit sphere endowed with the canonical and deformed Sasakian structures defined by Tanno. Then, composing with the flow of the Reeb vector field, we transform a biharmonic integral submanifold into a biharmonic anti-invariant submanifold. Using this method we obtain new examples of biharmonic submanifolds in spheres and, in particular, in $\mathbb{S}^{7}$.