类时Hilbert几何与Funk几何
Timelike Hilbert and Funk geometries
摘要：
类时空间是配备有偏序关系$<$和距离函数$ρ$的Hausdorff拓扑空间，所述距离函数满足公理集合，所述公理集合包括偏序关系和距离函数之间的一组相容性条件。距离函数仅定义在包含对角线的空间与其自身的乘积的子集上，即定义$ρ(x,y)$当且仅当$x<y$或$x=y$。三重$x,y,z$中不同点对之间的距离，无论何时定义这些距离，都满足所谓的\emph{time inequality}，这是一个反三角不等式$ρ(x,y)+ρ(y,z)\leq ρ(z,y)$。在1960年代，赫伯特·布斯曼（Herbert Busemann）发展了一个关于类时空间和局部类时空间的公理化理论。他的动机来自几何基础的相对论，他试图适应这一设置他的几何理论的度量空间，即他的理论$G$ -空间（测地空间）。他考虑的经典例子是$n$维洛伦兹空间。其他两个有趣的类的例子，他介绍了类时空间的类似物的芬克和希尔伯特几何。在本文中，我们调查这两个几何，并在这样做，我们介绍他们的变种，特别是类时相对芬克和希尔伯特几何，在欧几里德和球面设置。我们描述的芬斯勒无穷小结构的每一个这些几何形状（芬斯勒结构的适当概念），我们显示之间的相互作用的欧几里得和类时球形几何。特别是，我们刻画的德西特几何作为一个特殊的情况下，类时球面希尔伯特几何。本文的最终版本将出现在微分几何及其应用。
英文摘要：
A timelike space is a Hausdorff topological space equipped with a partial order relation $<$ and a distance function $ρ$ satisfying a collection of axioms including a set of compatibility conditions between the partial order relation and the distance function. The distance function is defined only on a subset of the product of the space with itself that contains the diagonal, namely, $ρ(x,y)$ is defined if and only if $x<y$ or $x=y$. Distances between pairs of distinct points in a triple $x,y,z$, whenever these distances are defined, satisfy the so-called \emph{time inequality}, which is a reverse triangle inequality $ρ(x,y)+ρ(y,z)\leq ρ(z,y)$. In the 1960s, Herbert Busemann developed an axiomatic theory of timelike spaces and of locally timelike spaces. His motivation comes from the geometry underlying the theory of relativity, and he tried to adapt to this setting his geometric theory of metric spaces, namely, his theory of $G$-spaces (geodesic spaces). The classical example he considers is the $n$-dimensional Lorentzian space. Two other interesting classes of examples of timelike spaces he introduced are the timelike analogues of the Funk and Hilbert geometries. In this paper, we investigate these two geometries, and in doing this, we introduce variants of them, in particular the timelike relative Funk and Hilbert geometries, in the Euclidean and spherical settings. We describe the Finsler infinitesimal structure of each of these geometries (with an appropriate notion of Finsler structure) and we display the interactions among the Euclidean and timelike spherical geometries. In particular, we characterize the de Sitter geometry as a special case of a timelike spherical Hilbert geometry. The final version of this paper will appear in Differential Geometry and its Applications.
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