Biharmonic curves in Finsler spaces
type: Item
instance of: scholarly article
zbMATH Open document ID: 1306.53028
DOI: 10.4134/JKMS.2014.51.6.1105
author: Nicoleta Voicu
published in: Journal of the Korean Mathematical Society
publication date: 2014-11-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1304.5499
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 58E10
Mathematics Subject Classification ID: 53C43
Mathematics Subject Classification ID: 53B40
Mathematics Subject Classification ID: 53C60
zbMATH DE Number: 6369558
zbMATH Keywords: bienergy functional
zbMATH Keywords: biharmonic curve
zbMATH Keywords: proper biharmonic curve
zbMATH Keywords: flag curvature
MaRDI profile type: Publication
arXiv ID: 1304.5499
Wikidata QID: Q125906767
title: Biharmonic curves in Finsler spaces
review text: The author extends the notion of bienergy functional to curves on Finsler spaces and deduces the equations of biharmonic curves in the Finslerian case. These equations involve, besides the flag curvature, two non-Riemannian objects: the Cartan tensor and the Landesberg tensor.NEWLINENEWLINEThe problem of existence of biharmonic curves is investigated and, as in Riemannian geometry, Finslerian biharmonic curves are shown to be of constant geodesic curvature. NEWLINENEWLINENEWLINEIt is found that all Finslerian geodesics are biharmonic but the converse is not true. For this converse, the author proves that any closed biharmonic curve is a geodesic. Moreover, in dimension two, for any Landsberg space with nonpositive flag curvature and any flat Finsler space (i.e., locally Minkowskian), biharmonic curves are geodesics. NEWLINENEWLINENEWLINEOn the other hand, the author shows that there exist biharmonic curves whish are not geodesics, namely, flat Finsler spaces of dimension higher than two. NEWLINENEWLINENEWLINEFinally, two concrete examples admitting proper biharmonic curves are given: a 2-dimensional projectively flat space and a 3-dimensional locally Minkowskian space.
reviewed by: Nabil L. Youssef
description: scientific article; zbMATH DE number 6369558