Bi-conformal vector fields and the local geometric characterization of conformally separable pseudo-Riemannian manifolds II

Details Bi-conformal vector fields and the local geometric characterization of conformally separable pseudo-Riemannian manifolds II Bi-conformal vector fields and the local geometric characterization of conformally separable pseudo-Riemannian manifolds II

2005-06-08

arxiv.org/abs/math/0506148v2

Mathematics

Differential Geometry

Minor changes, new references and an example added. 31 pages, no figures. To appear in Journal of Geometry and Physics

Scientific paper

In this paper we continue the study of bi-conformal vector fields started in {\em Class. Quantum Grav.} {\bf 21} 2153-2177. These are vector fields defined on a pseudo-Riemannian manifold by the differential conditions $\lie P_{ab}=\phi P_{ab}$, $\lie\Pi_{ab}=\chi\Pi_{ab}$ where $P_{ab}$, $\Pi_{ab}$ are orthogonal and complementary projectors with respect to the metric tensor $\rmg_{ab}$ and $\lie$ is the Lie derivative. In a previous paper we explained how the analysis of these differential conditions enabled us to derive local geometric characterizations of the most relevant cases of {\em conformally separable} (also called double twisted) pseudo Riemannian manifolds. In this paper we carry on this analysis further and provide local invariant characterizations of conformally separable pseudo-Riemannian manifolds with {\em conformally flat} leaf metrics. These characterizations are rather similar to that existing for conformally flat pseudo-Riemannian manifolds but instead of the Weyl tensor, we must demand the vanishing of certain four rank tensors constructed from the curvature of an affine, non-metric, connection (bi-conformal connection). We also speculate with possible applications to finding results for the existence of foliations by conformally flat hypersurfaces in any pseudo-Riemannian manifold.

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