AI 中文总结
本文研究不定统计流形光锥超曲面的径向与屏幕分布平行条件，给出基于屏幕形状算子和差分张量的判据，并探讨度量情形下联络与曲率性质。
AI 中文摘要
我们研究了不定统计流形中光锥超曲面相对于由环境对偶仿射联络诱导的联络的径向分布与屏幕分布。获得了每个分布相对于两个诱导联络均为平行的充分必要条件。这些条件通过屏幕形状算子、局部屏幕基本形式、平均诱导联络以及切向差分张量来表达。当两个分布均平行时，我们刻画了屏幕投影，并获得了切向差分张量的块分解，其中混合径向-屏幕部分消失。我们还获得了诱导联络成为度量联络的等价条件。在度量情形下，诱导的屏幕联络与屏幕分布每个积分流形上的Levi-Civita联络一致，并且三个诱导联络的曲率张量在屏幕方向上一致。我们进一步确定了这些曲率张量在径向方向上的作用。一个显式例子表明，两个分布可能均平行，尽管两个诱导联络都不是度量联络，且它们的曲率张量不必消失。
英文摘要
We study the radical and screen distributions of a lightlike hypersurface of an indefinite statistical manifold with respect to the induced connections arising from the ambient dual affine connections. Necessary and sufficient conditions are obtained for each distribution to be parallel with respect to both induced connections. These conditions are expressed through the screen shape operators, the local screen fundamental forms, the mean induced connection, and the tangential difference tensor. When both distributions are parallel, we characterize the screen projection and obtain a block decomposition of the tangential difference tensor, whose mixed radical-screen part vanishes. We also obtain equivalent conditions for the induced connections to be metric connections. In the metric case, the induced screen connections coincide with the Levi--Civita connection on each integral manifold of the screen distribution, and the curvature tensors of the three induced connections agree on screen directions. We further determine the action of these curvature tensors on the radical direction. An explicit example shows that both distributions may be parallel although neither induced connection is a metric connection and their curvature tensors need not vanish.
Comments11 pages
