Weakly conformal Finsler geometry
type: Item
instance of: scholarly article
title: Weakly conformal Finsler geometry
zbMATH Open document ID: 1311.53060
published in: Mathematische Nachrichten
publication date: 2014-11-12T00:00:00Z
Mathematics Subject Classification ID: 53C60
zbMATH DE Number: 6368941
zbMATH Keywords: Finsler metric
zbMATH Keywords: Randers metric
zbMATH Keywords: conformal geometry
author: Mehdi Rafie-Rad
MaRDI profile type: Publication
full work available at URL: mana.201300099
OpenAlex ID: W1943687895
cites work: ON THE FLAG CURVATURE OF FINSLER METRICS OF SCALAR CURVATURE
cites work: The action of conformal transformations on a Riemannian manifold
cites work: Weak Lie symmetry and extended Lie algebra
cites work: Homogeneous Riemannian spaces of four dimensions
cites work: The Obata conjecture for dimension 2
cites work: Lichnerowicz-Obata conjecture in dimension two
cites work: Finsler conformal Lichnerowicz-Obata conjecture
cites work: The Binet-Legendre metric in Finsler geometry
cites work: Transformation groups of spheres
cites work: The conjectures on conformal transformations of Riemannian manifolds
cites work: On conformal vector fields on Randers manifolds
cites work: On Finsler Spaces with Completely Integrable Equations of Killing
DOI: 10.1002/MANA.201300099
review text: In this paper, an extension of conformal equivalence for Finsler metrics is introduced and called weakly conformal equivalence and used to define weakly conformal transformations. The conformal Lichnerowicz-Obata conjecture is refined to weakly conformal Finsler geometry. It is proved that: If \(X\) is a weakly conformal complete vector field on a connected Finsler space \((M, F)\) of dimension \( n\geq 2 \), then, at least one of the following statements holds: (1) There exists a Finsler metric \( F_{1} \) weakly conformally equivalent to \(F\) such that \(X\) is a Killing vector field of the Finsler metric \( F_{1} \). (2) \(M\) is diffeomorphic to the sphere \(S^{n} \) and the Finsler metric is weakly conformally equivalent to the standard Riemannian metric on \( S^{n} \), and (3) \(M\) is diffeomorphic to the Euclidean space \(\mathbb R^{n} \) and the Finsler metric \(F\) is weakly conformally equivalent to a Minkowski metric on \(\mathbb R^{n} \). The vector field \(X\) with respect to the Minkowski metric is homothetic.NEWLINENEWLINENEWLINEIt is also proved that if the algebra of weak Killing vector fields of \(M\) has dimension more than \( \frac{n\left( n-1\right) }{2}+1 \) then, \(F\) is weakly isometric to a Riemannian metric of constant sectional curvature, where \(F\) is a Finsler metric on an \(n\)-dimensional manifold \( (n\neq 2,4) \).
reviewed by: Gauree Shanker
description: scientific article; zbMATH DE number 6368941