标题： 二维余维类空子流形与脐状类光法向截面及其与类光超曲面的关系 显示英文标题

标题： Codimension-Two Spacelike Submanifolds with Umbilical Lightlike Normal Sections and Their Relationship to Lightlike Hypersurfaces

摘要： 我们研究在允许脐状类光法向方向的洛伦兹时空中的余维二类空子流形。 我们证明这样的子流形受到强烈的几何和拓扑约束，建立了外在几何、平均曲率和剪切各向同性之间的明确关系。 在紧致情况下，我们获得了它们拓扑的严格限制。 我们精确地描述了何时可以将脐状类光法向向量场重新缩放为平行的，这取决于环境时空的曲率张量，并证明了这一性质是共形不变的。 我们的主要结果是一个分解定理：任何这样的子流形都包含在一个类光超曲面中，当类光法向方向是脐状时，该类光超曲面是完全脐状的。 我们还提供了由类光法向流生成的类空叶族上诱导度量之间的显式共形关系，这对等距性、类光法向方向的平行性、体积演化和变分性质有影响。 这些结果为广义相对论中视界和类光结构的数学研究提供了详细的几何框架。 显示英文摘要

摘要： We study codimension-two spacelike submanifolds in Lorentzian spacetimes that admit umbilical lightlike normal directions. We show that such submanifolds are subject to strong geometric and topological constraints, establishing explicit relationships between extrinsic geometry, mean curvature, and shear-isotropy. In the compact case, we obtain sharp restrictions on their topology. We precisely characterize when the umbilical lightlike normal vector field can be rescaled to be parallel, in terms of the curvature tensor of the ambient spacetime, and prove that this property is conformally invariant. Our main result is a factorization theorem: any such submanifold is contained in a lightlike hypersurface, which is totally umbilical whenever the lightlike normal direction is umbilical. We also provide explicit conformal relations between the induced metrics on the family of spacelike leaves generated by the lightlike normal flow, with consequences for isometry, parallelism of lightlike normal directions, volume evolution, and variational properties. These results yield a detailed geometric framework relevant to the mathematical study of horizons and lightlike structures in general relativity.