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

This is the first of two companion papers in which a thorough study of the normal form and the first integrability conditions arising from bi-conformal vector fields is presented. These new symmetry transformations were introduced in Class. Quantum Grav. 21, 2153–2177 and some of their basic properties were addressed there. Bi-conformal vector fields are defined on a pseudo-Riemannian manifold V   through the differential conditions £ξ→Pab=ϕPab and £ξ→Πab=χΠab where PabPab and ΠabΠab are orthogonal and complementary projectors with respect to the metric tensor gabgab. In our calculations a new affine connection (bi-conformal connection) arises quite naturally and this connection enables us to find a local characterization of conformally separable   pseudo-Riemannian manifolds (also called double twisted products) in terms of the vanishing of a rank three tensor TabcTabc. Similar local characterizations are found for the most important particular cases such as (double) warped products, twisted products and conformally reducible spaces.