Comment on Ricci Collineations for Type B Warped Space-times

We present two counter examples to the paper by Carot et al. in Gen. Rel. Grav. (1997). 29 1223 and show that the results obtained are correct but not general. (x α)) are a pair of pseudo-Riemannian manifolds with coordinate functions x A (A, B, ... = 1, 2) and x α (α, β, ... = 3, 4) respectively. Let Φ(x C) be a real valued function on M 1 and M = M 1 × M 2 be the product manifold (type B warped space-time) with metric [1]: ds 2 = h AB (x C)dx A dx B + Φ 2 (x C)h αβ (x γ)dx α dx β. A vector field X in M can be decomposed uniquely in " horizontal " and " vertical " components as follows: X a = X A 1 (x b)δ a A + X α 2 (x b)δ a α (2) where a, b, ... = 1, 2, 3, 4. In a recent paper [2] (referred from now on as CNP) Carot et al. have considered the problem of determining all Ricci collineations (RCs) of type B warped space-times and have come to the following conclusion: The horizontal component X 2 of a proper RC X in a warped type B space-time is either a Homothetic Vector Field (HVF) of (M 2 , h 2) and X is given by: