关于(α,β)-度量的S-曲率

On the S-curvature of Some (α,β)-metrics

给出(α,β)-度量F=αφ(β／α)的S-曲率的计算公式．证得对一般的(α,β)-度量,当β为关于α长度恒定的Nilling1-形式时,S=0.研究了Matsumoto-度量F=α2／(α-β)和(α,β)-度量F=α+∈β+k(β2／α)的S-曲率,证得S=0)当且仅当β为关于α长度恒定的Killing1-形式．同时还得到这两类度量成为弱Berwald度量的充要条件．其中φ(s)为光滑函数,为黎曼度量,β(y)=bi(x)yi为非零1-形式且∈,k≠0为常数．

This paper gives an explicit formula of the S-curvature of （α,β）-metrics F = αФ（β/α）, and proves that if β is Killing 1-form of constant length with respect to α, then S = 0. Next, the author studies the S-curvature of Matsumoto-metric F =α^2/（α - β） and （α,β）-metrics F = α ＋ εβ＋ κ（β^2/α）, and obtains that S = 0 if and only if β is Killing 1-form of constant length with respect to α. Actually, the author also obtains the condition of above two metrics to be weak Berwaldian. Here Ф（s） is a C^∞ function, α（y） = √√ij（x）y^iy^3 is Riemannian metric, β（y） = bi（x）y^i is non zero 1-form and ε, k ≠ 0 are constants.