Some Properties Of A Class Of Randers Metrics Is Pointwise Projective To Riemannian Metric

In this paper, we study some conditions and properties for an important class of randers metrics in forms of F(y)=(β(y)²+(1-|β|²)α(y)²)^1/2-β(y)/1-|β|² to be pointwiseprojective toα,whereα(y)=(α_ij(x)yⁱy^j)^1/2 is a Riemannian metric on ann-dimensional manifold M andβ(y)=b_i(x)yⁱ is a 1-form on M.We characterize some important conditions for F to be pointwise projective toαand discuss the conditions that F is Einstein metric or F is of constant flag curvature under the assumption that F is pointwise projective toα.Firstly, we study the condition that F is pointwise projective toαand obtain the following Theorem.Theorem 3.1. F is pointwinse projective toαif and only ifβsatisfies thefollowing equation Where (?)(y)=β(y)/1-|β|²,x(x)=1/1-|β|²,"|" denotes the covariant derivative with respcettoα.Furthermore , we discuss the conditions that F is Einstein metric under the assumption that F is pointwise projective toαand obtain the following Theorem.Theorem3.2 Assume that F is pointwise projective toαandαis anEinstein metric with Einstein constant (?) (that is , (?)(y) = (n -1)(?)α(y)²). Then Ric(y) =(n-1)λF(y)² if and only ifWhereΩ=Φ_|jy^j=(?)_l|k|jy^ly^kyj,ω=φ_|jy^j=x(|k|j)y^ky^j.Finally, we discuss the conditions that F is of constant flag curvature under the assumption that F is pointwise projective toαand obtain the following Theorem.Theorem 3.3 If F is pointwise projective toαandβsatisfies the following equationwhereμ= cons tan t. Then(i)Ifαis Ricci-flat ((?)=0),then F is an Einstein metric with Ric(y) = -(n - 1)μ²;(ii)Ifαis R-flat ((?)=0),then F is of constant curvature with R = -μ².