标题： 双共形矢量场及其在伪黎曼流形的保形可分离性刻画中的应用：伪黎曼流形中保形平坦叶层存在的新标准 显示英文标题

标题： Bi-conformal vector fields and their applications to the characterization of conformally separable pseudo-Riemannian manifolds: New criteria for the existence of conformally flat foliations in pseudo-Riemannian manifolds

摘要： 本文对由{\em 双保形矢量场}引发的法向形式和首次可积性条件进行了全面研究。 这些新的对称变换在{\em 类。 量子 引力。}\textbf{21} , 2153-2177 中被引入，并且在该文献中讨论了其中的一些基本性质。 双共形矢量场通过伪黎曼流形上的微分条件 $\lie P_{ab}=\phi P_{ab}$ 和 $\lie\Pi_{ab}=\chi\Pi_{ab}$ 定义，其中 $P_{ab}$ 和 $\Pi_{ab}$ 是相对于度规张量 $\rmg_{ab}$ 的正交且互补的投影算子。 我们研究的主要成果之一是发现了类似共形平坦度量的 Weyl 张量消失的新几何特征的 {\em 具有共形平坦叶度量的共形可分空间} 几何刻画。 这种几何特征似乎可以推广到任何允许共形平坦叶层的伪黎曼流形，这将为在给定的伪黎曼度规中系统寻找这类叶层打开大门。 其他相关方面，例如有限群生成元下不变张量的存在性等问题也得到了讨论。 显示英文摘要

摘要： In this paper a thorough study of the normal form and the first integrability conditions arising from {\em bi-conformal vector fields} is presented. These new symmetry transformations were introduced in {\em Class. Quantum Grav.}\textbf{21}, 2153-2177 and some of their basic properties were addressed there. Bi-conformal vector fields are defined on a pseudo-Riemannian manifold through the differential conditions $\lie P_{ab}=\phi P_{ab}$ and $\lie\Pi_{ab}=\chi\Pi_{ab}$ where $P_{ab}$ and $\Pi_{ab}$ are orthogonal and complementary projectors with respect to the metric tensor $\rmg_{ab}$. One of the main results of our study is the discovery of a new geometric characterization of {\em conformally separable spaces with conformally flat leaf metrics} similar to the vanishing of the Weyl tensor for conformally flat metrics. This geometric characterization seems to carry over to any pseudo-Riemannian manifold admitting conformally flat foliations which would open the door to the systematic searching of these type of foliations in a given pseudo-Riemannian metric. Other relevant aspects such as the existence of invariant tensors under the finite groups generated by these transformations are also addressed.