O’Neill’s book on Semi-Riemannian geometry has:
However, another book “pseudo-riemannian geometry δ-invariants and applications” has
Disagreeing on the sign of a term.
Which book has the correct formula? Are there any other authoritative sources?
Edit: A third source with a third set of signs! The book “Einstein Manifolds” has:
I've gone through and checked all three sources. O'Neill and Einstein Manifolds use a different sign convention from "Pseudo-riemannian Geometry" for the Riemann tensor, but they all define the Ricci tensor consistently. As mentioned above, O'Neill uses a positive sign for the Laplacian, but the other two use negative signs.
I checked the formula in O'Neill and verified that it is correct. "Einstein Manifolds" doesn't present its formulas for the Reimann curvature of a warped manifold, so I didn't check it, but it is consistent with O'Neill, taking into account the different Laplacian sign convention.
Finally, I checked "Pseudo-riemannian Geometry", and it seems to have an error; ${}^F Ric(V, W) - \ldots$ should be ${}^F Ric(V, W) + \ldots$:
$$ \begin{align*} \operatorname{Ric}(V, W) &= \sum_l \varepsilon_l \langle R(e_l, V)W, e_l \rangle \\ &= \sum_{e_l \in B} \varepsilon_l \langle R(e_l, V)W, e_l \rangle + \sum_{e_l \in F} \varepsilon_l \langle R(e_l, V)W, e_l \rangle \\ &= -(\langle V, W \rangle/f) \sum_{e_l \in B} \varepsilon_l \langle \nabla_{e_l} (\nabla f), e_l \rangle + \sum_{e_l \in F} \varepsilon_l \langle {}^F R(e_l, V) W, e_l \rangle + \frac{\langle \nabla f, \nabla f \rangle}{f^2} \sum_{e_l \in F} \varepsilon_l \langle \langle e_l, W\rangle V - \langle V, W \rangle e_l, e_l \rangle \\ &= \langle V, W \rangle (\Delta f/f) + {}^F\operatorname{Ric}(V, W) + \frac{\langle \nabla f, \nabla f \rangle}{f^2} \langle V, W \rangle (1 - k)\text{.} \end{align*} $$
