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.
Categories: math.DG gr-qc math-ph math.MP