广义半黎曼浸没与叶状结构
Generalized Semi-Riemannian Submersions and Foliations
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中文总结 AI 辅助
本文引入广义半黎曼浸没与叶状结构概念，建立正则叶状结构为广义半黎曼的充要条件，证明平稳横截叶状结构的诱导性质，得到时空上平稳类光叶状结构的刚性结果，并推导奥尼尔型截面曲率公式。
中文摘要 AI 辅助
本文引入广义半黎曼浸没与叶状结构的概念，将经典框架扩展至容纳具有变化或退化因果特征的叶，例如半黎曼流形上的齐次叶状结构，以及洛伦兹流形（包括pp-波）上的余维1类光叶状结构。我们证明，具有基本水平分布的正则叶状结构是广义半黎曼的，当且仅当它是横截的。此外，我们引入平稳叶状结构，并证明平稳横截叶状结构会在共形观测者的空间静止空间上诱导出经典黎曼叶状结构。作为几何推论，我们得到一个刚性结果，排除了正曲率罗伯逊-沃尔克时空或静态时空上的平稳余维1类光叶状结构。最后，我们对具有对合全分布的浸没推导了奥尼尔型截面曲率公式。
英文摘要
In this article, we introduce the concepts of generalized semi-Riemannian submersions and foliations, extending the classical framework to accommodate leaves with varying or degenerate causal characters, such as homogeneous foliations on semi-Riemannian manifolds and codimension-one lightlike foliations on Lorentz manifolds (including pp-waves). We establish that a regular foliation with a basic horizontal distribution is generalized semi-Riemannian if and only if it is transnormal. Furthermore, we introduce stationary foliations and prove that a stationary transnormal foliation induces a classical Riemannian foliation on the spatial rest space of a conformal observer. As a geometric consequence, we obtain a rigidity result ruling out stationary codimension-one lightlike foliations on positively curved Robertson-Walker spacetimes or static spacetimes. Finally, we derive an O'Neill-type sectional curvature formula for submersions with an involutive total distribution.
发表机构
- Universidade Federal da Bahia（巴伊亚联邦大学）
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