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

In this paper we continue the study of bi-conformal vector fields started in [A. García-Parrado, J.M.M. Senovilla, Class. Quant. Grav. 21 (2004) 2153–2177]. These are vector fields defined on a pseudo-Riemannian manifold by the differential conditions £ξ→Pab=ϕPab,£ξ→Πab=χΠab, where Pab,ΠabPab,Πab are orthogonal and complementary projectors with respect to the metric tensor gabgab. 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 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 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.