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Classification of biharmonic Riemannian submersions from manifolds with constant sectional curvature

Shun Maeta, Miho Shito

Abstract

In 2011, Wang and Ou (Math. Z. 269:917-925, 2011) showed that any biharmonic Riemannian submersion from a 3-dimensional Riemannian manifold with constant sectional curvature to a surface is harmonic. In this paper, we generalize the 3-dimensional setting to arbitrary dimensions. By constructing an adapted orthonormal frame, we simplify the biharmonic equation for Riemannian submersions and analyze the curvature properties of Riemannian manifolds with constant sectional curvature. As a result, we prove that a Riemannian submersion from an $(n+1)$-dimensional Riemannian manifold with constant sectional curvature to an $n$-dimensional Riemannian manifold is biharmonic if and only if it is harmonic.

Classification of biharmonic Riemannian submersions from manifolds with constant sectional curvature

Abstract

In 2011, Wang and Ou (Math. Z. 269:917-925, 2011) showed that any biharmonic Riemannian submersion from a 3-dimensional Riemannian manifold with constant sectional curvature to a surface is harmonic. In this paper, we generalize the 3-dimensional setting to arbitrary dimensions. By constructing an adapted orthonormal frame, we simplify the biharmonic equation for Riemannian submersions and analyze the curvature properties of Riemannian manifolds with constant sectional curvature. As a result, we prove that a Riemannian submersion from an -dimensional Riemannian manifold with constant sectional curvature to an -dimensional Riemannian manifold is biharmonic if and only if it is harmonic.

Paper Structure

This paper contains 3 sections, 6 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the Main Theorem and Supporting Lemmas

Key Result

Theorem 1.1

Let $\phi : (M^{n+1}(c), g) \longrightarrow (N^n, h)$ be a Riemannian submersion from a Riemannian manifold with constant sectional curvature c. Then, $\phi$ is biharmonic if and only if it is harmonic.

Theorems & Definitions (14)

  • Conjecture 1: Chen's conjecture C
  • Conjecture 2: Generalized Chen's conjecture CMO
  • Conjecture 3: BMO conjecture BMO
  • Theorem 1.1
  • Theorem 2.1: Theorem 3.1 in AO
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 4 more