On (α, β)-metrics With K=1 And S=0

In this paper, we study the S-curvature and flag curvature of (α,β)-metrics. First, we calculate the Busemann-Hausdorff volume and give explicit formula of the S-curvature for (α,β)-metrics. Then we find a sufficient condition for (α,β)-metrics of vanished Scurvature. And we really find a class of Finsler metrics on Lie group S³ with vanished curvature. Intrigued by D.Bao and Z. Shen's example, we calculate the flag curvature of (α,β)-metrics under the condition thatβis a Killing 1-form of constant length with respect toα, and then we find a class of Finsler metrics with constant flag curavature K=1 among the above metrics.Proposition For (α,β)-metrics F =αφ(β/α), Let dV_F =σF(χ)ω¹âˆ§â€¦âˆ§Ï‰ⁿ and dV_α=σ_α(χ)ω¹âˆ§â€¦Ï‰ⁿ be the Busemann-Haudorff volume forms of F and a respectively,thenσ_F(χ) =μ(b)σα(χ), whereμ(b) = andΓ(χ) is Eulerfunction.Theorem 1- For (α,β)-metrics F =αφ(β/α), ifβis a Killing 1-form of constant with respect toα, then S = 0.Example^[BS]: Letακ=(k²|θ¹|²+k|θ²|²+k|θ³|²)^1/2 be Riemannian metrics andβ_κ=(k²-kθ¹)^1/2 be 1-form globally defined on the Lie group S³, then a easy culculation givesthatβ_κis Killing 1-form with respect toα_κand ||β_κ||α_κ= (k²-k)^1/2/k= constant. By theorem 1 we have that the S-curvature of F_κ=α_κφ(β_κ/α_κ) vanishes. And in [BS], if F =α_κ+β_κ, the flag curvature of F_κis K = 1. We generalize this result:Theorem 2: Letα_κ=(k2[θ¹]² + k[θ²]² + k[θ³]^21/2 be Riemannian metrics andβ_κ=(k²-kθ¹)^1/2 be 1-form globally defined on the Lie group S³. Then has the flag curvature K = 1, whereλ:= (k-1)/k and k≥1, c1≤k are constants. We study some special (α,β)-metrics F=∈β+κ(β²/α) and Matsumoto-metrics F =α²/(α-β) and obtain the following resultTheorem 3: For Matsumoto-metrics F =α²/(α-β) and (α,β)-metrics F =α+∈β+κ(β²/α), where∈,κare non-zero numbers. Then the following are equivalent:(1) F has isotropic S-curvature, i. e. there exists a scalar function c(x) on M such that S = (n + 1)c(x)F.(2) F has isotropic mean Berwald-curvature, i. e. there exists a scalar function c(x) on M such that E = (n+1)/2c(x)F^-1h.(3)βis Killing 1-form of constant length with respect toα, i. e.γ₀₀= 0, s₀ = 0.(4) S = 0.(5) F is weak-Berwaldian, i. e. E = 0.