论文总字数：34892字
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\begin{center}{\songti \zihao{2}{芬斯勒流形上一类共形变换的研究}}\end{center}
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\begin{center}{\songti \zihao{4} 摘\ \ \ \ 要}
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\addcontentsline{toc}{chapter}{摘\ \ \ \ 要} {\songti\zihao{-4} \ \
本文首先介绍了芬斯勒流形的度量结构，由此结构引出流形上丰富多彩的黎曼型和非黎曼型几何量，初步计算了一些几何量共形变换下的表达式。在此基础上，进一步研究在共形变换时S 曲率，E 曲率，$\Xi$ 曲率的变换性质。最后我们转向特殊的Randers 度量空间来计算几何量的表达式，以及S 曲率的共形关系式。
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\vskip 1cm \noindent{\songti \zihao{-4}关键词: \ 芬斯勒流形,\ 共形变换, \ 曲率量, \ Randers度量 }
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\begin{center}{ \rm A class of conformal transformations on Finsler manifolds}
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\begin{center}{ \rm Abstract}
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This paper first introduces the metric structure of the Finsler manifold, and the structure leads to the rich and colorful Riemannian and non-Riemannian geometric quantities on the manifold, and the expressions under some geometrical conformal transformations are preliminarily calculated. On this basis, the transformation properties of S curvature, E curvature, and $\Xi$ curvature in the conformal transformation are further studied. Finally, we turn to the special Randers metric space to calculate the expression of the geometric quantity, and the conformal relationship of the S curvature.
\vskip 0.8cm \noindent{\rm Key Words:\ Finsler manifolds, \ conformal transformation, \ curvature, \ Randers metric}
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%东~南~大~学~毕~业~设~计~报~告}\hspace{0.5cm}{\bf 第一章 \qquad 引言}
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\chapter{介绍}
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%芬斯勒结构，共形变换已有的工作。
芬斯勒流形是黎曼流形的推广。在黎曼的博士论文中，他已经认识到芬斯勒流形的重要性，并选择了特殊的二次型的度量，也就是黎曼度量，从而给出了相应的黎曼结构。而芬斯勒结构则更加一般，蕴含的内容也更加丰富。
\begin{defi}
设M是一个n维$ C^\infty $ 流形，TM是M的切丛。M的一个（全局定义的）芬斯勒结构是一个满足下列性质的函数\[ F:\,TM\rightarrow [0,\infty) \]
(i) 正则性：在穿孔切丛$ TM\backslash \{0\} $ 上，F是 $ C^\infty $ 的；\\
(ii) 正齐次性：对$ \forall \lambda gt; 0 $，$ F(x,\lambda y) = \lambda F(x,y) $；\\
(iii)强凸型: $ n\times n $黑塞矩阵\[
(g_{ij}):=\left( [\frac12F^2]_{y^iy^j} \right)
\]
在穿孔切丛$ TM\backslash 0 $上每一点都是正定的。
\end{defi}
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