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$f$-Biharmonic submanifolds in space forms and $f$-biharmonic Riemannian submersions from 3-manifolds

Ze-Ping Wang, Li-Hua Qin

Abstract

$f$-Biharmonic maps are generalizations of harmonic maps and biharmonic maps. In this paper, we obtain some descriptions of $f$-biharmonic curves in a space form. We also obtain a complete classification of proper $f$-biharmonic isometric immersions of a developable surface in $\r^3$ by proving that a proper $f$-biharmonic developable surface exists only in the case where the surface is a cylinder. Based on this, we show that a proper biharmonic conformal immersion of a developable surface into $\r^3$ exists only in the case when the surface is a cylinder. Riemannian submersions can be viewed as the dual notion of isometric immersions (i.e., submanifolds). We also study $f$-biharmonicity of Riemannian submersions from 3-space forms by using the integrability data. Examples are given of proper $f$-biharmonic Riemannian submersions and $f$-biharmonic surfaces and curves.

$f$-Biharmonic submanifolds in space forms and $f$-biharmonic Riemannian submersions from 3-manifolds

Abstract

-Biharmonic maps are generalizations of harmonic maps and biharmonic maps. In this paper, we obtain some descriptions of -biharmonic curves in a space form. We also obtain a complete classification of proper -biharmonic isometric immersions of a developable surface in by proving that a proper -biharmonic developable surface exists only in the case where the surface is a cylinder. Based on this, we show that a proper biharmonic conformal immersion of a developable surface into exists only in the case when the surface is a cylinder. Riemannian submersions can be viewed as the dual notion of isometric immersions (i.e., submanifolds). We also study -biharmonicity of Riemannian submersions from 3-space forms by using the integrability data. Examples are given of proper -biharmonic Riemannian submersions and -biharmonic surfaces and curves.
Paper Structure (6 sections, 20 theorems, 68 equations)

This paper contains 6 sections, 20 theorems, 68 equations.

Table of Contents

  1. Introduction and Preliminaries
  2. $f$-Biharmonic submanifolds in space forms
  3. $f$-Biharmonic curves in space forms
  4. $f$-Biharmonic isometric immersions and biharmonic conformal immersions of a developable surface into $\hbox{${\mathbb R}$}^3$
  5. $f$-biharmonic Riemannian submersions from 3-manifolds
  6. $f$-biharmonic Riemannian submersions from $3$-space forms

Key Result

Lemma 2.1

Let $\gamma: (a,b)\to(N^{n}, h) (n\geq 2)$ be a curve parametrized by arc length into a Riemannian manifold. Then $\gamma$ is $f$-biharmonic if and only if :

Theorems & Definitions (44)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Remark 1
  • Theorem 2.4
  • proof
  • Remark 2
  • Corollary 2.5
  • ...and 34 more