摘要:
Finsler几何是没有二次型限制的黎曼几何, 在Finsler几何中很重要的两个问题是射影平坦和对偶平坦的Finsler度量.本文主要研究了一类含有3个参数的Finsler度量F\!=\!\alpha+\beta,其中 alpha(x,y)\!=\!\frac{\sqrt{\kappa^2{\langlex,y\rangle}^2+\varepsilon{\mid y\mid}^2(1+\zeta{\midx\mid}^2)}}{1+\zeta{\midx\mid}^2和beta(x,y)=\frac{\kappa\langlex,y\rangle}{1+\zeta{\mid x\mid}^2}$.利用Hamel方程和对偶平坦方程,得到了这类Finsler度量为射影平坦和对偶平坦的充要条件.
Abstract:
Finsler geometry is just Riemannian geometry without quadratic restriction, and we know that the projectively flat and dually flat Finsler metrics are two of important problems in Finsler geometry. In this paper, we study a class of Finsler metrics with 3 parameters in the form $F=\alpha+\beta$, where $\alpha(x,y)=\frac{\sqrt{\kappa^2{\langle x,y\rangle}^2+\varepsilon{\mid y\mid}^2(1+\zeta{\mid x\mid}^2)}}{1+\zeta{\mid x\mid}^2}$ and
$\beta(x,y)=\frac{\kappa\langle x,y\rangle}{1+\zeta{\mid x\mid}^2}$.By using the Hamel's equations and dually flat equations, the necessary and sufficient conditions for the Finsler metrics to be projectively flat and dually flat are obtained.
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