On a class of Finsler metrics of scalar flag curvature
type: Item
instance of: scholarly article
title: On a class of Finsler metrics of scalar flag curvature
zbMATH Open document ID: 1359.53015
author: Qiaoling Xia
published in: Results in Mathematics
publication date: 2017-02-22T00:00:00Z
review text: Let \(M\) be a smooth manifold of dimension \(n\). A Finsler metric having the form \(F =\alpha\phi(b^2, s);\,\, s:= \frac{\beta}{\alpha}\), is called a general \((\alpha,\beta)\)-metric, where \(\alpha= \sqrt {a_{ij}(x)y^iy^j}\) is a Riemannian metric, \(\beta = b_i(x)y^i\) is a 1-form on \(M\) and \(b:= b(x) = ||\beta(x)||_\alpha\) is the norm of \(\beta\) with respect to \(\alpha\). In particular, if \(\phi_1 = 0\), the metric \(F\) reduces to the usual \((\alpha,\beta)\)-metric \(F = \alpha\phi(s)\). A Finsler metric \(F\) on \(M\) is said to be locally projectively flat if at any point of \(M\), there is a local coordinate system in which \(F\) is projectively flat. It is known that every locally projectively flat Finsler metric is of scalar flag curvature. However, the converse is not true. The author of the present paper is concerned with this converse. He tries to characterize Finsler metrics of scalar (resp. constant) flag curvature for a certain class of general \((\alpha,\beta)\)-metrics. He proves that a class of general \((\alpha,\beta)\)-metrics satisfying certain conditions is of scalar flag curvature if and only if it is locally projectively flat. Moreover, he finds a system of nonlinear partial differential equations characterizing the above class of general \((\alpha,\beta)\)-metrics such that it is of scalar (resp. constant) flag curvature and determines its local structure by solving these equations. As an application, the author obtains some non-projectively flat Finsler metrics of scalar (resp. constant) flag curvature.
Mathematics Subject Classification ID: 53B40
Mathematics Subject Classification ID: 53C60
zbMATH DE Number: 6687990
zbMATH Keywords: general \((\alpha,\beta)\)-metric
zbMATH Keywords: locally projectively flat
zbMATH Keywords: flag curvature
zbMATH Keywords: Douglas metric
reviewed by: Nabil L. Youssef
MaRDI profile type: Publication
full work available at URL: s00025-016-0539-6
OpenAlex ID: W2281298264
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Wikidata QID: Q126013373
DOI: 10.1007/S00025-016-0539-6
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description: scientific article; zbMATH DE number 6687990