Triharmonic Riemannian submersions from 3-dimensional manifolds of constant curvature

TL;DR

This paper proves that triharmonic Riemannian submersions from 3D space forms to surfaces are necessarily harmonic, advancing the understanding of biharmonic map conjectures.

Contribution

It provides a partial affirmative answer to the submersion version of the generalized Chen's conjecture for triharmonic maps.

Findings

Triharmonic Riemannian submersions from 3D space forms are harmonic.

Supports the conjecture that higher-order harmonic submersions are harmonic.

Extends previous results on biharmonic maps to triharmonic cases.

Abstract

For biharmonic maps, there is a famous conjecture named Chen's conjecture. In later paper, Wang and Ou gave an affirmative partial answer to submersion version of Chen's conjecture. In this paper, we give an affirmative partial answer to submersion version of generalized Chen's conjecture, that is, triharmonic Riemannian submersions from a 3-dimensional space form into a surface is harmonic.