具有可反测地线奇异的(α,β)度量
On Singular (α,β)-Metrics with Reversible Geodesics
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摘要: 如果Finsler空间中的测地线沿相同的路径反向也是一条测地线，则称该Finsler空间具有可反的测地线.当流形维数n>2时，已刻画了具有可反测地线的（α，β）-度量，但是并没有考虑奇异的情形.无论奇异与否，当流形维数n>2时，给出了具有可反测地线的（α，β）-空间的分类.进一步，也证明了可反的Finsler空间在进行了Kropina变换之后仍具有可反的测地线.
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关键词:
- 奇异的(α,β)-度量 /
- Kropina变换 /
- 可反的测地线
Abstract: A Finsler space is said to have reversible geodesics if for any of its oriented geodesic paths, the same path traversed in the opposite sense is also a geodesic. I M Masca, S V Sabau and H Shimada characterize the (α,β)-metrics with reversible geodesics when the dimension n>2, but they do not think about the singular (α,β)-metrics. In this paper, whether (α,β)-metrics are singular or not, we provides a complete classification of them with reversible geodesics when the dimension n>2. Further, we prove the Kropina change of a reversible Finsler metric must have reversible geodesics.-
Key words:
- Singular (α,β)-Metrics /
- Kropina Change /
- Reversible Geodesic .
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[1] BRYANT R.Geodesically Reversible Finsler 2-Spheres on Constant Curvature[J]. GriffithsNankai Tracts in Math,2006, 11:95-111. [2] CRAMPIN M. Randers Spaces with Reversible Geodesics[J]. PublMath Debrecen, 2005, 67(3/4):401-409. [3] MASCA IM, SABAU S V,SHIMADA H.Reversible Geodesics for (α, β)-Metrics[J].InternatJMath, 2010, 21(8):1071-1094. [4] CHERN SS, SHEN Z. Rieman-Finsler Geometry[J].World Scientific,2005, 1(4):485-498. [5] MASCA IM, SABAU S V,SHIMADA H.Two-Dimensional (α, β)-Metrics with Reversible Geodesics[J].PublMathDebrecen, 2013, 82(2):485-501. -
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