On \(\eta\)-biharmonic hypersurfaces in pseudo-Riemannian space forms
type: Item
instance of: scholarly article
title: On \(\eta\)-biharmonic hypersurfaces in pseudo-Riemannian space forms
author: Li Du
author: Jinjun Ren
published in: Mathematica Slovaca
publication date: 2022-10-19T00:00:00Z
review text: An \(\eta\)-biharmonic submanifold is an isometric immersion \(\phi: (M^m, g)\to (N,h)\) which is a critical point of the linear combination \(E_2(\phi)+\eta E_1(\phi)\) of the energy and the bienergy functionals. The equation for an \(\eta\)-biharmonic submanifold can be described as: \(\tau_1(\phi)=m\eta H\), where \(H\) is the mean curvature and \(\tau_2(\phi)\) is the bitension field of the submanifold. The paper under review studies \(\eta\)-biharmonic hypersurfaces with constant scalar curvature in 5-dimensional pseudo-Riemannian space forms. Among other things, the authors of the paper proved that if the shape operator of such hypersurface is diagonalizable, then the \(\eta\)-biharmonic hypersurface has constant mean curvature. Some partial classification of such hypersurfaces are also given.
reviewed by: Ye-Lin Ou
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 7604041
zbMATH Keywords: pseudo-Riemannian space forms
zbMATH Keywords: \(\eta\)-biharmonic hypersurfaces
zbMATH Keywords: diagonalizable shape operator
zbMATH Keywords: constant scalar curvature
MaRDI profile type: Publication
full work available at URL: ms-2022-0086
OpenAlex ID: W4312842460
cites work: Q3788641
cites work: Hypersurfaces of \(E_s^4\) with proper mean curvature vector
cites work: Hypersurfaces of type \(M^3_2\) in \(E^4_2\) with proper mean curvature vector
cites work: Lorentz hypersurfaces in \(E_{1}^{4}\) satisfying \(\Delta\overset\rightarrow H=\alpha \overset\rightarrow H\)
cites work: Null 2-type surfaces in \(E^ 3\) are circular cylinders
cites work: Q3974897
cites work: Q4883994
cites work: Q3070798
cites work: BIHARMONIC SURFACES IN PSEUDO-EUCLIDEAN SPACES
cites work: BIHARMONIC PSEUDO-RIEMANNIAN SUBMANIFOLDS IN PSEUDO-EUCLIDEAN SPACES
cites work: Hypersurfaces of \(\bar\mathbb{E}^ 4\) satisfying \(\Delta\vec H=\lambda\vec H\)
cites work: Biharmonic submanifolds of pseudo-Riemannian manifolds
cites work: Classification of \(\eta \)-biharmonic surfaces in non-flat Lorentz space forms
cites work: Biharmonic submanifolds with parallel normalized mean curvature vector field in pseudo-Riemannian space forms
cites work: Hypersurfaces satisfying \(\tau_2(\phi)=\eta\tau (\phi)\) in pseudo-Riemannian space forms
cites work: On surfaces in the 3-dimensional Lorentz-Minkowski space
cites work: Classifying hypersurfaces in the Lorentz-Minkowski space with a characteristic eigenvector
cites work: Biharmonic hypersurfaces with three distinct principal curvatures in Euclidean 5-space
cites work: On biharmonic hypersurfaces with constant scalar curvatures in $\mathbb S^5$
cites work: Q5440873
cites work: Q3729625
cites work: Q3764911
cites work: Q4138831
cites work: Q5460094
cites work: Classification of proper biharmonic hypersurfaces in pseudo-Riemannian space forms
cites work: Hypersurfaces in \(\mathbb E_s^{n+1}\) satisfying \(\Delta\overrightarrow H=\lambda\overrightarrow H\) with at most three distinct principal curvatures
cites work: Lorentz hypersurfaces in \(\mathbb{E}_1^{n + 1}\) satisfying \({\Delta} \overrightarrow{H} = \lambda \overrightarrow{H}\) with at most three distinct principal curvatures
cites work: Hypersurfaces in \(\mathbb{E}_s^{n + 1}\) satisfying \(\operatorname{\Delta} \overrightarrow{H} = \lambda \overrightarrow{H}\) with at most two distinct principal curvatures
cites work: Q3312869
cites work: Some recent progress of biharmonic submanifolds
cites work: Biharmonic submanifolds in nonflat Lorentz 3-space forms
DOI: 10.1515/MS-2022-0086
description: scientific article; zbMATH DE number 7604041