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 关于局部对偶平坦的 Douglas\! \mbox{\boldmath $(\alpha,\beta)$}-}度量 On Locally Dually Flat Douglas \mbox{\boldmath $(\alpha,\beta)$}-metrics 蒋经农1, 程新跃2, 田艳芳3 JIANG Jingnong1,*, CHENG Xinyue2, TIAN Yanfang3 1.遵义医学院医学信息工程系, 遵义,贵州, 563003;2. 重庆理工大学数学与统计学院,重庆, 400054; 3. 解放军后勤工程学院基础部, 重庆, 400016 1. Department of Medical Information Engineering, Zunyi Medical College, Zunyi, Guizhou, 563003, P. R. China; 2. School of Mathematics and Statistics, Chongqing University of Technology, Chongqing, 400054, P. R. China;3. Department of Fundamental Courses, 收稿日期: 2011-08-23 出版日期: 2013-10-25 DOI: 10.11845/sxjz.20130516
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 摘要 本文证明了如果形如 $F=\alpha \phi(\frac{\beta}{\alpha})$的多项式$(\alpha, \beta)$-度量为局部对偶平坦的Douglas度量，则此度量一定是Berwald度量, 其中$\phi(s)$ 是一个关于 s 的 k 阶$(k\geq2)$ 多项式, $\alpha$是一个黎曼度量, $\beta$是一个非零的1-形式. 特别地, 在$\phi^{\prime}(0)$满足特定条件的情形下, 得到了这类多项式$(\alpha, \beta)$-度量为局部对偶平坦的Douglas度量的充要条件. 关键词 ： 芬斯勒度量,  $(\alpha,\beta)$-度量,  局部对偶平坦的$(\alpha,\beta)$-度量,  Douglas度量 Abstract：In this paper, we prove thatifthe polynomial $(\alpha,\beta)$-metrics in the form of $F=\alpha \phi(\frac{\beta}{\alpha})$ arelocally dually flat Douglas metrics, then they must be Berwald metrics, where $\phi(s)$ is a polynomial in s of degree $k\ (k\geq 2)$, $\alpha$ is a Riemannian metric and $\beta$ is a 1-form. In particular, we obtain sufficient and necessary conditions for such polynomial $(\alpha,\beta)$-metrics to be locally dually flat Douglas metrics under special conditions about $\phi^{\prime}(0)$. Key words： $(\alpha,\beta)$-metric    locally dually flat $(\alpha,\beta)$-metric    Douglas metric 基金资助:Foundation item:Supported by NSFC (No. 10971239) and Natural Science Foundation of Guizhou Province (No. JLKZ201201)
 [1] 郑大小,贺群. 具有近迷向旗曲率\ \mbox{\boldmath$K=\frac{3c_{x^i}y^i}{F}+\sigma$}的\mbox{\boldmath$(\alpha, \beta)$}度量[J]. 数学进展, 2015, 44(4): 599-606. [2] 蒋经农，程新跃. 两类重要的(α,β)-度量之间的射影变换(英)[J]. 数学进展, 2012, 41(6): 732-740.
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