论文标题
灯光歧管和卡坦几何形状
Lightlike manifolds and Cartan geometries
论文作者
论文摘要
引入了灯泡的彩色几何形状，作为cartan几何形状，以洛伦兹 - 米科夫斯基时空的未来灯泡锥进行建模。然后，我们从这个角度提供了一种研究光歧管的方法。据说，歧管$ n $上的每个灯具纸箱几何形状都提供了灯光级的$ h $，其根本分布在全球范围内由矢量场$ z $跨越。对于Lorentz歧管的浅色超曲面，我们给出的条件是在环境歧管的Levi-Civita连接形式的下拉形式的倾斜上，是这种超曲面上的灯塔连接。在特殊的情况下，灯光般的超曲面完全完全脐带，这种结构本质上返回原始的轻度度量。从固有的角度来看，从给定的灯光歧管$（N，H）$开始，我们展示了一种构建一个环境洛伦兹歧管系列的方法，这些方法将$（n，h）$视为hypersurface。此方法的灵感是在同形几何形状的Feffermann-Graham环境构造上的灵感，并在$（n，h）$是通用时在原始歧管上提供了灯泡的cartan几何形状。
Lightlike Cartan geometries are introduced as Cartan geometries modelled on the future lightlike cone in Lorentz-Minkowski spacetime. Then, we provide an approach to the study of lightlike manifolds from this point of view. It is stated that every lightlike Cartan geometry on a manifold $N$ provides a lightlike metric $h$ with radical distribution globally spanned by a vector field $Z$. For lightlike hypersurfaces of a Lorentz manifold, we give the condition that characterizes when the pull-back of the Levi-Civita connection form of the ambient manifold is a lightlike Cartan connection on such hypersurface. In the particular case that a lightlike hypersurface is properly totally umbilical, this construction essentially returns the original lightlike metric. From the intrinsic point of view, starting from a given lightlike manifold $(N,h)$, we show a method to construct a family of ambient Lorentzian manifolds that realize $(N,h)$ as a hypersurface. This method is inspired on the Feffermann-Graham ambient metric construction in conformal geometry and provides a lightlike Cartan geometry on the original manifold when $(N,h)$ is generic.
