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On (α, β)-metrics with K=1 and S=0

Author: CuiNingWei
Tutor: WangJia
School: Southwestern University
Course: Basic mathematics
Keywords: Finlser geometry S- curvature (α, β) - metrics Flag curvature
CLC: O186.1
Type: Master's thesis
Year: 2007
Downloads: 25
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Abstract


(Α, β) - S-curvature metric and flag curvature. First, by calculation Busemann-Hausdorff volume form, gives the formula for the calculation of the S-curvature. To thereby obtain a (α, β) - measure of curvature of the S-0 in a non-trivial condition. And found that the existence of a group in the Lie group S 3 Riemannian metric and the overall definition of the 1 - form by any smooth function φ constructed (α, β) - metrics have S = 0. Inspired β Killing constant α length 1 - form the flag curvature calculation, construct a Lie group S 3 with K = 1 and S = 0 of a Group Finsler metric. Specifically, the following results: Proposition: (α, β) - metric F = αφ (β / α), so the dV F = σ (x) ω F (x) ω , 1 ∧ ... ∧ ω n and dV α = σ α ∧ ... ∧ ω n F and α Busemann-Hausdorff volume form: σ F (x) = μ (b) σ α (x), wherein, Γ (x) is the Euler-function. Theorem 1: (α, β) - metric F = αφ (β / α), when β the the α constant length Killing1-form, S = 0. Examples [BS] : Let Riemannian metric on Lie group S 3 , β k = (k 2 -k) 1/2 θ 1 1 in the form of the overall definition of the Lie group S 3 . Can be obtained by calculating: β k the form of Killing1-Riemannian metric α k and satisfies ‖ β k the ‖ of αk < / sub> = ((k 2 -k) 1/2 ) / k is a constant. By the Theorem 1 the known F k = α k φ the (β k / α k ) S-curvature for 0. We generalize this conclusion: Theorem 2: Let the Riemannian metric on the Lie group S 3 , β k = (k 2 -k ) 1/2 θ 1 1 in the form of the overall definition of the Lie group S 3 . Flag curvature K = 1 and S = 0, where λ: = (k-1) / k, k ≥ 1, c 1 ≤ k is an arbitrary constant. Theorem 1 converse proposition is established, we have a special metric F = α ∈ β k (β 2 / α) and Matsumoto-metric F = α 2 / ( α-β) do attempt to obtain: Theorem 3: the Matsumoto-metric F = α 2 / (alpha-beta), and (alpha, beta) - metric F = α ∈ β k (β 2 / α), where ∈, k is nonzero constant. The following equivalence: (1) F having a fan, i.e. there is M, the number of function c (x), such that S = (N 1) c (x) is F to the S-curvature. (2) F having a fan to the average curvature of the Berwald-, i.e. M is the number of the function C (x), such that E = (n 1) / 2C (x) F -1 h. (3) the β α constant length Killing1-form, i.e. r the 00 = 0, S 0 = 0. (4) S = 0. (5) F weak Berwald metric, i.e. E = 0.

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CLC: > Mathematical sciences and chemical > Mathematics > Geometry, topology > Differential geometry,integral geometry > Differential Geometry
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