Randers metrics with special curvature properties
type: Item
instance of: scholarly article
title: Randers metrics with special curvature properties
zbMATH Open document ID: 1100.53056
author: Xinyue Chen
author: Zhongmin Shen
published in: Osaka Journal of Mathematics
publication date: 2003-11-03T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1153493038
review text: In Finsler geometry, besides the flag curvature \(\mathbb K\) there exist several non-Riemannian quantities: \(\tau\) (distortion),\(\mathbb I\) (the mean Cartan torsion), \(\mathbb S\) (the S--curvature), \(\mathbb E\) (the E--curvature), \(\mathbb J\) (the mean Landsberg cuvature) cf. [\textit{Zhongmin Shen}, Differential geometry of spray and Finsler spaces. Dordrecht: Kluwer Academic Publishers (2001; Zbl 1009.53004)]. The authors study these quantities for the Randers metrics and prove Theorem 1.1: Let \(F = \alpha + \beta \) be a Randers metric on an \(n\)- dimensional manifold \(M\). (i) For a scalar function \(c=c(x)\) on \(M\), the following are equivalent: a) \({\mathbb S}= (n+1)cF\); b)\({\mathbb E} = (1/2)(n+1)cF^{-1}h\), where \(h\) is the angular metric; (ii) For a scalar function \(c=c(x)\) on \(M\), the following are equivalent: \({\mathbb J} +cF{\mathbb I}=0;{\mathbb S} = (n+1)cF \) and \(\beta\) is closed. Then they classify the Randers metrics with constant flag curvature \({\mathbb K} = \lambda\) and \({\mathbb J}/I = -c(x)F \).
Mathematics Subject Classification ID: 53C60
zbMATH DE Number: 1894628
zbMATH Keywords: Randers metrics
zbMATH Keywords: curvatures
reviewed by: Mihai Anastasiei
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1894628