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

In this paper a thorough study of the normal form and the first integrabilityconditions arising from {\em bi-conformal vector fields} is presented. Thesenew symmetry transformations were introduced in {\em Class. QuantumGrav.}\textbf{21}, 2153-2177 and some of their basic properties were addressedthere. Bi-conformal vector fields are defined on a pseudo-Riemannian manifoldthrough the differential conditions $\lie P_{ab}=\phi P_{ab}$ and$\lie\Pi_{ab}=\chi\Pi_{ab}$ where $P_{ab}$ and $\Pi_{ab}$ are orthogonal andcomplementary projectors with respect to the metric tensor $\rmg_{ab}$. One ofthe main results of our study is the discovery of a new geometriccharacterization of {\em conformally separable spaces with conformally flatleaf metrics} similar to the vanishing of the Weyl tensor for conformally flatmetrics. This geometric characterization seems to carry over to anypseudo-Riemannian manifold admitting conformally flat foliations which wouldopen the door to the systematic searching of these type of foliations in agiven pseudo-Riemannian metric. Other relevant aspects such as the existence ofinvariant tensors under the finite groups generated by these transformationsare also addressed.