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

Abstract

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 → P a b = ϕ P a b and £ ξ → Π a b = χ Π a b where P a b and Π a b are orthogonal and complementary projectors with respect to the metric tensor g a b . 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 T a b c . Similar local characterizations are found for the most important particular cases such as (double) warped products, twisted products and conformally reducible spaces.