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\begin{document}
\title{Erratum to ``A Berger-Green type inequality for compact Lorentzian manifolds"}
\author{Manuel Guti\'{e}rrez, Francisco J. Palomo}
\author{Alfonso Romero}
\subjclass{53C50, 53C22.}

\address{Departamento de \'Algebra,
Geometr\'{\i}a y Topolog\'{\i}a. Facultad de Ciencias, Universidad de
M\'alaga, Campus Teatinos, 29071- M\'alaga, Spain.}
\email{M.G: {\ttfamily mgl@agt.cie.uma.es}\\
F.J.P: {\ttfamily fjpalomo@eresmas.com}}
\address{Departamento de Geometr\'{\i}a y Topolog\'{\i}a, Universidad
de Granada, 18071-Granada, Spain.}
\email{{\ttfamily aromero@ugr.es}}

\maketitle

The results of the paper \cite{GutPalRom01} remains true as stated, but there is a mistake in the proof of Proposition 2.3, namely the assertion that the direct sum decomposition (2.1) is an orthogonal decomposition. This  fact has been also used in the proofs of Proposition 2.4 and Theorem 2.6. The gaps may be amended as follows. \vspace{1mm}


{\it Proof of Proposition 2.3}.
Let $\mathbf{V} \in \mathfrak{X}(TM)$ be given by $\hat{g}(\mathbf{V}_{v},X)=g(c(X),K_{\pi v})+g(v,\nabla_{d\pi (X)}K)$, for every $X\in T_{v}TM$. Thus we have ${\hat g}(\mathbf {V}_v,{\mathbf A}_v)=1$, at
any $v\in C_K(M)$, and 
so $D_v=\mathrm {Span}\{{\mathbf V}_v,{\mathbf A}_v\}$ is
a Lorentzian vector subspace of $T_vTM$ (recall that 
${\hat g}({\mathbf A},{\mathbf A})=0$ on $C_{K}M$). One easily checks that $X\in T_{v}C_{K}M$ if and only if
$\hat{g}(\mathbf{A}_{v},X)=\hat{g}(\mathbf{V}_{v},X)=0$. Therefore, $(C_{K}M,\hat{g})$ is a Lorentzian manifold. The proof for spacelike fibres remains valid.

Finally, 
$
[ T_{v}(C_{K}M)_{p}] ^{\perp }=\{X\in T_{v}TM:c(X)=-g(v,\nabla_{d\pi (X)}K)\, v\},
$
and so $\pi$ restricted to $C_{K}M$ is a semi-riemannian submersion.
\qed \vspace{1mm}


{\it Proof of Proposition 2.4}.
For every $v \in C_{K}M$, we put $\mathbf{K}_{v}=\mathbf{K}^{T}_{v}+\mathbf{K}^{N}_{v}$, with $\mathbf{K}^{T}_{v}\in T_{v}C_{K}M$ and $\mathbf{K}^{N}_{v}\in D_{v}$. Let $m\in \N$ be such that $\mathbf{K}^{m}=\mathbf{K}^{N}-m\mathbf{A}$ is a timelike vector field on an open neighbourhood $O\subset C_{K}M$ of $v$. Taking into account that $i_{\mathbf{K}^{m}}i_{\mathbf{A}}\Omega=i_{\mathbf{K}}i_{\mathbf{A}}\Omega$ on $C_{K}M$, we replace $\mathbf{K}$ in (2.8) by $\mathbf{K}^{m}$ and proceed as there to get the result on each $O$.
\qed \vspace{1mm}

The paragraph after Proposition 2.4 (Since...result.) must be deleted. \vspace{1mm}

{\it Proof of Theorem 2.6}.
The proof remains valid until $c(X)=...=\frac{\nabla J}{dt}\mid_{0}$. The remainder must be modified as follows. Thus, (2.12) and (2.11) yield $\mathfrak{L}_{Z_{g}}\beta_{g}(X)=(\rho \circ \pi)(v)\alpha_{g}(X)-2\sigma_{g}(X)$ where $\sigma_{g}(X)=g(c(X),K_{\pi v})$. But $\sigma_{g}(Z_{g})=0$, and since for any $v\in C_{K}M$, $0\not= Z_{g}(v)\in T_{v}C_{K}M$ and both $\alpha_{g}(Z_{g})$, $i_{Z_{g}}(d\alpha_{g})$ vanish on $C_{K}M$, the result easily follows from (2.10).
\qed
\begin{thebibliography}{99}
\bibitem{GutPalRom01}  {\sc M. Guti\'errez, F.J. Palomo and A. Romero},
A Berger-Green type inequality for compact Lorentzian manifolds, {\sl Trans. Amer.
Math. Soc.~\bf 354} (2002) 4505-4523.
\end{thebibliography}
\end{document}