标题： 各向异性共形变换的锥面伪芬斯勒曲面，I 显示英文标题

标题： Anisotropic conformal change of conic pseudo-Finsler surfaces, I

摘要： 本工作旨在研究圆锥伪芬斯勒曲面 $(M,F)$的各向异性共形变换，即 $ F(x,y)\longmapsto \overline{F}(x,y)=e^{\phi(x,y)}F(x,y)$，其中函数 $\phi(x,y)$依赖于位置 $x$和方向 $y$，与仅依赖于位置的普通（各向同性）共形变换相反。 如果 $F$是一个伪芬斯勒度量，则上述变换不一定产生一个伪芬斯勒度量。 因此，我们找到了一个(圆锥)伪芬斯勒曲面$(M,F)$在变换$\overline{F}=e^{\phi(x,y)}F$下转化为另一个(圆锥)伪芬斯勒曲面$(M,\overline{F})$的充分必要条件。 在一般维度下，以张量形式找到逆度量张量的各向异性共形变化是非常困难的。 然而，通过使用芬斯勒曲面上的修改后的伯尔瓦德标架，我们得到了逆度量张量分量的张量形式的变化。 这一进展使我们能够研究芬斯勒几何对象的变换以及与变换后的芬斯勒函数$\overline{F}$相关的几何性质。 与各向同性共形变换不同，我们有一个非等距共形因子$\phi(x,y)$，它保持测地喷射不变。 此外，我们还发现了一些在各向异性共形变换下保持不变的几何对象。 此外，我们研究了$\overline{F}$成为对偶平坦和/或射影平坦的充分条件。 最后，我们研究了共形因子$\phi(x,y)$的一些特殊情况。 在需要的时候，提供了各种例子。 显示英文摘要

摘要： The present work is devoted to investigate anisotropic conformal transformation of conic pseudo-Finsler surfaces $(M,F)$, that is, $ F(x,y)\longmapsto \overline{F}(x,y)=e^{\phi(x,y)}F(x,y)$, where the function $\phi(x,y)$ depends on both position $x$ and direction $y$, contrary to the ordinary (isotropic) conformal transformation which depends on position only. If $F$ is a pseudo-Finsler metric, the above transformation does not yield necessarily a pseudo-Finsler metric. Consequently, we find out necessary and sufficient condition for a (conic) pseudo-Finsler surface $(M,F)$ to be transformed to a (conic) pseudo-Finsler surface $(M,\overline{F})$ under the transformation $\overline{F}=e^{\phi(x,y)}F$. In general dimension, it is extremely difficult to find the anisotropic conformal change of the inverse metric tensor in a tensorial form. However, by using the modified Berwald frame on a Finsler surface, we obtain the change of the components of the inverse metric tensor in a tensorial form. This progress enables us to study the transformation of the Finslerian geometric objects and the geometric properties associated with the transformed Finsler function $\overline{F}$. In contrast to isotropic conformal transformation, we have a non-homothetic conformal factor $\phi(x,y)$ that preserves the geodesic spray. Also, we find out some invariant geometric objects under the anisotropic conformal change. Furthermore, we investigate a sufficient condition for $\overline{F}$ to be dually flat or/and projectively flat. Finally, we study some special cases of the conformal factor $\phi(x,y)$. Various examples are provided whenever the situation needs.