标题： 关于共形流形的马科维茨伪距离 显示英文标题

标题： On Markowitz's pseudodistance for conformal manifolds

摘要： 在20世纪80年代，M. J. Markowitz在伪黎曼流形上引入了一种共形不变的伪距离，其灵感来自于射影几何中的Kobayashi度量。 这一构造依赖于一类特殊的参数化类光测地线，称为射影参数化。 我们首先回顾这种伪距离的基本性质，并提供几个例子族，在这些例子中该伪距离是非退化的，并且在某些情况下是完备的。 特别是，我们研究了三类流形：闭合流形、爱因斯坦宇宙中的共形凸区域以及全局双曲、共形平坦、$C$-极大时空。 对于前两类流形，我们得到了类似于Brody和Barth关于复Kobayashi度量的结果。 最后，我们将Markowitz的伪距离应用于分类爱因斯坦-德西特空间的所有准齐次区域，即由类空超平面限定的闵可夫斯基空间的一个半空间。 在共形变换下，仅存在有限多个这样的区域，而且所有这些区域都证明是齐次的。 显示英文摘要

摘要： In the 1980s, M. J. Markowitz introduced a conformally invariant pseudodistance on pseudo-Riemannian manifolds, inspired by the Kobayashi metric in projective geometry. This construction relies on a distinguished class of parametrized lightlike geodesics, called projectively parametrized. We begin by reviewing the fundamental properties of this pseudodistance and provide several families of examples where it is non-degenerate and, in some cases, complete. In particular, we investigate three classes of manifolds: closed manifolds, conformally convex domains of the Einstein universe, and globally hyperbolic, conformally flat, $C$-maximal spacetimes. For the first two classes, we obtain results analogous to those of Brody and Barth concerning the complex Kobayashi metric. Finally, we apply Markowitz's pseudodistance to classify all quasi-homogeneous domains of the Einstein-de Sitter space, that is, a half-space of Minkowski space bounded by a spacelike hyperplane. Up to conformal transformations, only finitely many such domains exist, and all of them turn out to be homogeneous.