Projective Change Between Two Special Classes Of (α, β)-Metrics

The （α, β）-metric is an important class of Finsler metrics including Randers metricas the simpliest class. Recently, the projective changes between these two special met-rics have been studied by many geometers. For example, in2011, M. Zohrehvand andM. M. Rezaii considered the projective change between a （α, β） metric F=α²/（α β）anda Randers metric（?）=（?）+（?）. In this paper, we will study projective change between a（α, β）-metric F=εβ+α+（3/2） arctan（β/α）+αβ²/2（α²+β²ï¼‰and Randers metric（?）=（?）+（?）.This paper is divided into three parts: In the first part, we introduce the researchbackground and the related primary theorems and definitions which prepare for thediscussed work. In the second part of the paper, firstly, we introduce the projective flatof （α, β）-metrics F=εβ+α++（3/2） arctan（β/α）+αβ²/2（α²+β²ï¼‰. Secondly, we give the relatedresults of projective change. My main results lie in the third part. We investigate theprojective change between F and Randers metric（?）.Our main results can be stated as follows:Lemma3.1: Let F=εβ+α+（3/2） arctan（β/α）+αβ²/2（α²+β²ï¼‰and（?）=（?）+（?） betwo （α, β）-metric on a manifold M with dimension n≥3, where α and （?） are twoRiemannian metrics, β and（?） are two nonzero one forms. Then they have the sameDouglas tensor if and only if both F and（?） are Douglas metrics.Theorem3.2: Let F=εβ+α+（3/2） arctan（β/α）+αβ²/2（α²+β²ï¼‰and（?）=（?）+（?）be two （α, β）-metric on a manifold M with dimension n≥3, where α and （?） are twoRiemannian metrics, β and（?） are two nonzero one forms. Then F is projectively relatedto（?） if and only if the following equations holdG_αⁱ=G_?ⁱ+θ_yⁱ-4τα²bⁱ,b_i|j=2Ï„[（1+4b²ï¼‰Î±_ij-3b_ib_j],d（?）=0where bⁱ:=a^ijb_j, b:=‖β‖α, b_i|j denote the covariant derivatives of β with respect toα, Ï„=τ（x） is a scalar function and θ=θ_iyⁱ is a one form on M.