# Biharmonic isometric immersions into and biharmonic Riemannian submersions from Berger 3-spheres

## TL;DR

This paper classifies proper biharmonic isometric immersions and Hopf tori in Berger 3-spheres, and shows that biharmonic Riemannian submersions from Berger spheres are necessarily harmonic.

## Contribution

It provides the first complete classification of proper biharmonic surfaces and submersions in Berger 3-spheres, revealing new geometric properties.

## Key findings

- Proper biharmonic surfaces with constant mean curvature are classified.
- Proper biharmonic Hopf tori are completely characterized.
- Biharmonic Riemannian submersions from Berger spheres are equivalent to harmonic ones.

## Abstract

In this paper, we study biharmonic isometric immersions of a surface into and biharmonic Riemannian submersion from 3-dimensional Berger spheres. We obtain a classification of proper biharmonic isometric immersions of a surface with constant mean curvature into Berger 3-spheres. We also give a complete classification of proper biharmonic Hopf tori in Berger 3-sphere. For Riemannian submersions, we prove that a Riemannian submersion from Berger 3-spheres into a surface is biharmonic if and only if it is harmonic.